What does SOHCAHTOA mean in trigonometry?

SOHCAHTOA stands for Sine, Cosine, and Tangent, with their ratios: Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, and Tangent is opposite over adjacent.

Where does trigonometry originate?

Trigonometry began with the study of the stars by ancient Babylonians, who used geometric concepts to map the night sky.

What are coterminal angles?

Coterminal angles share the same terminal arm in standard position. They can be found by adding or subtracting full circles (360 degrees) to the original angle.

What is an angle in standard position?

Standard position means the angle's initial arm is on the positive x-axis, with the vertex at the origin and the terminal arm anywhere on the Cartesian plane.

What are the primary trigonometric ratios?

Primary trigonometric ratios are sine, cosine, and tangent, which relate the sides of a right triangle.

Math 20-1 - Lesson 7.1 - intro to trigonometry

00:33:10

https://www.youtube.com/watch?v=fUkzfUFhYOo

Summary

TLDRThe session introduced the new unit on trigonometry for this class. Key topics included reviewing Grade 10 concepts like SOHCAHTOA and understanding their application in labeling right triangles. Historical context was provided, tracing trigonometry's roots to ancient Babylonian studies of the stars using circle geometry. The lesson highlighted the primary trigonometric ratios: sine, cosine, and tangent, explaining them through right triangles. Furthermore, the concept of angles in standard position was introduced on the Cartesian plane, including how to label them and understand coterminal angles through rotational symmetry on the circle. Practical tips such as ensuring calculators are in the correct mode (degree) and problem-solving approaches for right triangle issues were emphasized.

Takeaways

🔢 SOHCAHTOA helps remember trigonometric ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
📜 Trigonometry originated with Babylonians studying the stars, using geometry to understand the sky.
🖊️ Label sides and angles of triangles using opposite and adjacent relations.
🧭 Angles in standard position have their initial arm on the positive x-axis.
🎯 Coterminal angles share the same final position on the plane, adding or subtracting 360° to find them.
🗺️ Understand angles with rotational symmetry using the Cartesian plane.
🔍 Right triangle problems can be solved with primary trigonometric ratios and the Pythagorean theorem.
📐 Ensure calculators are set in degree mode for trigonometric calculations.
💡 Angles less than 90° are acute and important in triangle calculations.
🔄 Reference angles are always formed with the x-axis in trigonometry.

Timeline

00:00:00 - 00:05:00
The lesson begins the final unit on trigonometry for the class, with a review of key concepts from Grade 10. The teacher emphasizes the importance of recalling the primary trigonometric ratios: sine, cosine, and tangent, remembered through the acronym SOHCAHTOA. This review sets the stage for the study of angles in standard positions.
00:05:00 - 00:10:00
The origins of trigonometry are explored, tracing back to ancient Babylon and the study of stars. The teacher explains the practical applications of trigonometry today, such as in GPS technology. The lesson then shifts to labeling triangles, defining angle types, and reviewing the Pythagorean theorem, reinforcing foundational knowledge.
00:10:00 - 00:15:00
Detailed instructions are provided for ensuring calculators are in degree mode for the calculations relevant to Grade 11. Students are reminded that errors in calculator mode can affect trigonometric functions but not basic arithmetic. Solving triangles involves determining unknown angles and sides, using learned ratios and the Pythagorean theorem.
00:15:00 - 00:20:00
Students practice solving right triangles by applying trigonometric identities and the Pythagorean theorem to find unknown angles and sides. Emphasis is placed on rounding results correctly and ensuring understanding of 'solving' as calculating all angles and sides from given information.
00:20:00 - 00:25:00
Introduction to angles in the standard position on the Cartesian plane, explaining the concepts of initial and terminal arms. Students learn to draw and interpret angles correctly in quadrants, and the significance of positive and negative angle rotations. Reference angles are introduced for understanding acute angles to the x-axis.
00:25:00 - 00:33:10
The concept of coterminal angles is addressed, explaining how an angle can share the same terminal point by rotating integer multiples of 360 degrees. Students learn to identify and describe these angles using equations, understanding the infinite possibilities of coterminal angles through real-world examples and practice problems.

Mind Map

Video Q&A

What does SOHCAHTOA mean in trigonometry?
SOHCAHTOA stands for Sine, Cosine, and Tangent, with their ratios: Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, and Tangent is opposite over adjacent.
Where does trigonometry originate?
Trigonometry began with the study of the stars by ancient Babylonians, who used geometric concepts to map the night sky.
What are coterminal angles?
Coterminal angles share the same terminal arm in standard position. They can be found by adding or subtracting full circles (360 degrees) to the original angle.
What is an angle in standard position?
Standard position means the angle's initial arm is on the positive x-axis, with the vertex at the origin and the terminal arm anywhere on the Cartesian plane.
What are the primary trigonometric ratios?
Primary trigonometric ratios are sine, cosine, and tangent, which relate the sides of a right triangle.

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Subtitles

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00:00:00
hey friends
00:00:01
so we're gonna start our new unit today
00:00:05
uh which is on trick now uh for those of
00:00:08
you who are with me at school
00:00:10
um this is our last unit of 20-1
00:00:13
if you are joining us from a different
00:00:16
school this may not be your last unit uh
00:00:19
but for us uh in my class this was how
00:00:22
we are going to end our time together so
00:00:26
pretty exciting uh you guys have come a
00:00:28
long way and you've journeyed
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really well
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so pat yourself on the back
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um okay so
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the purpose of the first half of this
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lesson is just to kind of review some of
00:00:39
the things we did in grade 10 in
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tennessee uh to make sure that you're
00:00:44
okay with everything make sure we're all
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kind of going um starting off on the
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same page okay um the back half of this
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lesson we're gonna talk about angles in
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standard position and kind of introduce
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that concept but first um we just need
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to go back to sokatoa make sure
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everybody's uh cool with that um most of
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you no matter what school you would have
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gone to uh in grade 10 you would have
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had a formula sheet for math 10c that
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would have included the primary
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trigonometric ratios so sine
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is opposite over hypotenuse cosine is
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adjacent over hypotenuse and tangent is
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opposite over
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adjacent
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those aren't given to you anymore in
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grade 11. um so
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a very helpful acronym is sohcahtoa see
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how
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uh neat i can make my mouse writing be
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today sokatoa now this is just a quick
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little acronym to help you remember
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um
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who goes
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ah that was really bad oh okay who goes
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with what okay so if i just start at the
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left here and read with me the s is for
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sign and then the first uh
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or the next letter would be the one on
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top of the fraction for the ratio and
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then the third letter would be the one
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on the bottom of the fraction for the
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ratio so sine is opposite over
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hypotenuse
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cosine is adjacent over hypotenuse and
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then tangent is opposite over adjacent
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okay so sohcahtoa just a quick way to
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help you remember
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okay
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so
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um
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let's give it a quick go here we're
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gonna start with a very brief history
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and won't take too long on this but i
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think it's important that you know where
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some of the stuff comes from okay
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let's begin with a history of
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trigonometry which doesn't begin
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actually with triangles it actually
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begins with circles the study of
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trigonometry dates back to ancient
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babylon we're talking around 2000 bc
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as scholars attempted to understand the
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motion of the stars across the night sky
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really a lot of trigonometry was birthed
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from them trying to chart these stars
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and trying to say okay if the stars are
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in this position where am i sitting on a
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boat in the ocean or where am i sitting
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on this land mass and that's how we
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actually started mapping out the world
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they conjectured that the stars lie on a
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great sphere about the earth
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and consequently assume that their
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motions across the sky were along great
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circular arcs
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a natural question might be where on
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this circle
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must the star be if i'm viewing it at a
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particular angle of x so you can kind of
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see a person there staring up at the
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star um and thinking about what angle
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that forms
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okay
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uh and that's this the uh study of
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circle geometry was born
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trigonometry is actually one of the most
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used mathematical concepts um ever okay
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um
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it's used a lot in trades it's used in
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programming it's used in
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triangulation of
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cell phone data like the very fact that
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your cell phone can pinpoint where you
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are in the world up to about a meter
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um it's because your cell phone connects
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to three different towers okay
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and then how um
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how it connects to those three tower
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towers how it forms that triangle
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actually
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allows satellites to figure out exactly
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where that cell phone is
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in the vast world that's pretty
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incredible when you stop and think about
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it
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okay so the first thing we want to do is
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just review how to label triangles so
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i've given you a right triangle here and
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you know it's right because of the
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box down here by a
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we label angles as capital letters and
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we put those capital letters just on the
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outside edge so inside here is angle a
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and so we put a capital a on the corner
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inside here is angle b so we have a
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capital b on the corner and inside here
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is angle c and we have a capital c
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on the corner okay we also have this
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symbol here this symbol is called theta
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um and it's just used often in math for
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an unknown angle okay
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um
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so theta often used for an angle we
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don't know the measure
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now
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what we want to do is we want to put
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some words on here and we want to label
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these guys with lower case letters okay
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and the way we do lower case letters is
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we do the same letters as the
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angles so
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what i want you to do is i want you to
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put your pencil on a
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and i want you to just go away from it
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okay
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that will point to little a
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okay when we say go away we're talking
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actually about opposites okay so we say
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opposite angle a is little a
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opposite angle b well i put my pen on b
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and i go away from it
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it will point to the opposite side of b
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so that would be little b right here and
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then for c i put my angle at c and i go
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away from it well it's going to point to
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little c right here
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okay
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then we also have words for these guys
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um this one here you should know because
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he's opposite the 90 degree angle and
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you were introduced to that concept um a
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little while ago probably back in grade
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8.
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um that's called the hypotenuse okay the
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hypotenuse in grade 8 you would have
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been introduced to the pythagorean
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theorem which says that the hypotenuse
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squared
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will equal the two legs squared and
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added together so the two other sides
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squared and added together okay
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now we also have words of opposite and
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adjacent that we need to talk about and
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they go depending on
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the non-ninety degree angle you're
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looking at so that's why i've stuck
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theta here
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okay
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um opposite theta is the opposite side
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okay
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and then
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so i've got the opposite i've got the
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hypotenuse over here i didn't actually
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label that sorry i talked about it but i
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didn't label it
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and then this guy here is adjacent
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adjacent means to
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be the side
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okay so the adjacent side is the one
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that helps form where that angle is but
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it's not the hypotenuse
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okay
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so now i'm going to clear that drawing
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and i'm just going to bring it a little
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neater
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okay here's everything i said there
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don't let me just go back for one sec
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sorry um
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previous okay and i'm just gonna hide my
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face here so you can see the word theta
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if you don't have the book at home
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okay
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awesome now
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um these are called the primary
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trigonometric ratios the sohcahtoa that
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i started you with
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the sine ratio the cosine ratio and the
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tangent ratio okay
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so the sine ratio is abbreviated sin on
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the on our calculator you'll see an sin
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button it's not a sin button we try not
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to sing
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it's a sine button so sine of theta is
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opposite over hypotenuse the cosine of
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theta is the ratio of the adjacent side
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over the hypotenuse side
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and then the tangent of theta is the
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ratio of the opposite divided by dg
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okay
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and the pythagorean theorem a squared is
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my hypotenuse for this guy be careful
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with the letters right the letters
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themselves don't matter
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what matters is whatever has been
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labeled as the hypotenuse that has to be
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by its side so it might be a in another
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situation it might be c then another
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situation might be w it doesn't really
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matter okay
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it's the hypotenuse squared
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equals the square of the other two legs
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the b and the c here
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um added together
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okay
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again i'm just gonna hide myself for a
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second so that you guys can see that at
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home
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um and then the other thing i need to
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remind you guys about is that you need
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to be in degree mode so there are
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two um
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ways we can measure
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just like if i showed you the messy desk
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that you can't see because it's right
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underneath the screen right now but i
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promise you it's messy there's piles of
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stuff everywhere um if i wanted to
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measure the length of my messy desk i
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have two options with which to measure
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it in i could measure it in centimeters
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or i could measure it in inches those
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would be two very different numbers but
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that doesn't change the length of my
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desk
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okay it's the same with angles when i go
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to measure an angle i have two different
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measurements i could use i could use or
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units of measure i should say i could
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use
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degrees or i could use radians now by
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default calculators graphing calculators
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are set in radians because radians is
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actually the much better mathematical
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um
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way to
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use the unit of measure for angles okay
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but we're not going to deal with that
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until grade 12. so for grade 11 we need
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to make sure your calculator is always
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in degree mode so that we can talk in
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degrees
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okay
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now to do that i've given some
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instructions here just to make sure
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everybody's good
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if you are a child of casio you'll go
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shift menu to get into your setup scroll
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down to angle and press f1 for degree
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if you are a child of ti
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you will press the mode button and
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go down three over one you'll notice if
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you go down three you have the words
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radian and degree and by default radian
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will be highlighted so then if you go
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over one
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um you'll be on the word degree and if
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you hit just hit enter it'll highlight
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degree instead
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as a check especially if you have one of
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these guys or if you have a scientific
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calculator and you want to make sure
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you're absolutely in the right mode 10
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45 should equal exactly one okay the tan
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of 45 should be one
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now uh kids get really uh stressed out
00:11:02
about
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modes
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um
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so just for clarification
00:11:08
the mode you're in doesn't matter unless
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you are pressing the sign the cosine or
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the tangent button okay
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no matter what mode you're in two plus
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two is going to equal four all the
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regular features of your calculator are
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going to work the only difference is
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when you go to press the sine the cosine
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or the tangent button okay
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all right now this phrase would have
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been introduced to you in uh grade 10 to
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solve a triangle means to determine the
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measure of the missing angles and the
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lengths of the missing sides
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in order to determine the unknown
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measures we use primary trigonometric
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ratios or the pythagorean theorem
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essentially
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um last year i could give you any right
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triangle
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there are six things is the first thing
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i need you to understand there right
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there's three angles and three sides so
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last year i could give you any right
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triangle um and i could give you three
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items so the 90 degree and maybe a side
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and an angle or the 90 degree and two
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sides or whatever
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and you could solve for the other three
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okay
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um this year we will actually be able to
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extend that to say it doesn't even have
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to be a 90 degree angle anymore there
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are other tools that we will learn in
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grade 11
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where i can give you any three items and
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you'll be able to get me the other three
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whether there's a 90 degree angle there
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or not
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okay
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all right so let's try a couple
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solve the following angle round side
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lengths to the nearest tenth and angles
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to the nearest degree it's super
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important that you pay attention to
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um
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rounding instructions okay
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side lengths have to be to the nearest
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tenth angles have to be to the nearest
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degree if you don't round this proper
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i'll take off half a point okay
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you have to follow the directions if you
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forget your units i'll take off half a
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point okay you have to talk about what
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unit of measure you're in so all of
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those things are super important okay
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so you have been given the 90 degree
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angle here you've been given little t
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here
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and you've been given
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a little w
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here
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so what you have to now do is find me
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angle f
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find me angle t
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and find me little f
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side f okay i'm gonna actually start
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with side f um i know
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two sides of a right triangle so i can
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get the third using the pythagorean
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theorem okay uh if i'm going to use the
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pythagorean theorem i need to think
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about the fact that i have the
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hypotenuse here so it's going to be f
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squared plus 7 squared equals 14 squared
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some of you from your junior high days
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may want to go directly to 14 squared
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minus 7 squared equals f squared that's
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totally fine as well okay
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so 14 squared equals seven and up seven
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squared plus f squared
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um so then
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14 squared is 196 7 squared is 49 i'll
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subtract 49 from both sides to get f
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squared and then the last step will be i
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need to uh square root it and follow the
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proper rounding construction so that's
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going to give me 12.1 centimeters okay
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12.1 centimeters for uh psi f
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now
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i could go ahead and solve for angle t
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or angle f
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if possible i'm not going to use this in
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my next step okay i could use it but
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what if i'm wrong
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if i'm wrong by default the next guy is
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going to be wrong right so try to only
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use what's been given to you
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uh whenever possible okay so if i wanted
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to get angle f
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okay the 7
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would be adjacent
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and
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i'm just going to put an a there
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and the 14
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would be
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the hypotenuse
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if i wanted to get angle
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t
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the 7
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would be my opposite
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yeah
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and the 14
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would be my hypotenuse okay so i'm going
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to set myself up
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to solve for f and to solve for angle t
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using
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cosine for angle f
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adjacent and hypotenuse that's what
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pairs adjacent and hypotenuse together
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is cosine
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and then sine for angle t because i need
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to pair up opposite end
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hypotenuse
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okay
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so i started with cos of f is 7 over 14.
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remember when i want to get f by itself
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i will arc cos or inverse cos this
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so f will be the inverse cose of one
00:15:46
half i just reduced my seven over
00:15:47
fourteen
00:15:48
okay
00:15:50
and then
00:15:51
that gives me sixty degrees then for t
00:15:55
i'm going to use sine
00:15:58
sine of t is 7 over 14.
00:16:01
so then i'm going to arc
00:16:03
sine
00:16:04
or
00:16:05
inverse sine one half and get 30 degrees
00:16:09
and of course as a check these two
00:16:12
numbers added together should give me 90
00:16:14
because i need 90 plus this 90 to get
00:16:17
the 180 that all
00:16:19
angles in a triangle have to add up to
00:16:21
okay
00:16:23
so
00:16:24
i'm on a good path here
00:16:26
um okay let me clear this ink and we'll
00:16:28
do another example together
00:16:32
so this time
00:16:34
i have triangle drw
00:16:36
where r angle r is 90 degrees
00:16:40
little r is 12
00:16:42
and the little w is 7. now the
00:16:44
orientation doesn't matter okay
00:16:46
you can orient orientate orient
00:16:49
orientate you can orientate this
00:16:52
triangle however you want okay
00:16:54
um
00:16:56
what matters is wherever you put that 90
00:16:59
degrees you have to label that as r
00:17:02
opposite that so the hypotenuse has to
00:17:04
be labeled as 12.
00:17:06
and then wherever you chose to write w
00:17:09
opposite w has to be a seven
00:17:11
okay
00:17:12
so
00:17:13
here's mine
00:17:15
but again you have you may have your
00:17:17
setup slightly different and that's
00:17:19
totally fine
00:17:20
okay
00:17:21
so now
00:17:22
i need to get
00:17:24
a little d
00:17:29
and i need to get angle w and i need to
00:17:31
get angle d
00:17:33
okay
00:17:34
so a little d i'm going to use the
00:17:35
pythagorean theorem again
00:17:41
so i'll go 12 squared minus 7 squared
00:17:43
essentially okay 12 squared's 144 7
00:17:47
squared's 49 i'll subtract the 49 from
00:17:49
both sides
00:17:52
and then i'll square root it i got to
00:17:53
follow rounding instructions this one
00:17:55
said all answers should be to one
00:17:57
decimal place so i've got 9.7
00:18:01
okay
00:18:02
now if i want angle w
00:18:06
um i'm going to be looking at opposite
00:18:10
and hypotenuse
00:18:13
so that's going to use sine
00:18:16
if i'm looking at angle d i'm going to
00:18:19
look at adjacent
00:18:23
and hypotenuse
00:18:25
so that's going to be cosine okay
00:18:31
so
00:18:32
i started with
00:18:34
angle d
00:18:35
cosine of d is 7 over 12 i'm going to
00:18:38
inverse cos or arc cos that's what gets
00:18:41
rid of the coast that's attached to the
00:18:43
d and allows d to be by itself
00:18:46
so arc cos 7 over 12 and that's going to
00:18:49
give me 54.3 again be careful with your
00:18:51
rounding instructions it said all
00:18:53
answers to one decimal place don't get
00:18:55
caught into thinking that
00:18:57
angles have to always be rounded to the
00:18:58
nearest whole number you just do
00:19:00
whatever the rounding instructions tell
00:19:01
you to do
00:19:03
okay
00:19:05
and then for angle w
00:19:07
we'll have sine of w is 7 over 12
00:19:11
um and then we'll arc sine
00:19:14
7 over 12
00:19:16
and get 35.7
00:19:19
again as a check
00:19:22
we know that
00:19:24
35.7
00:19:26
plus
00:19:27
the
00:19:28
54.3
00:19:29
should give me 19. okay
00:19:32
these two here should give me 90 and
00:19:34
then when i add that to the other 90
00:19:36
over here
00:19:38
i'll get 180
00:19:40
okay
00:19:43
awesome
00:19:46
okay
00:19:48
so the triangles we had above i'm just
00:19:51
going to kill my face here for sex so
00:19:53
that you guys can read this with me the
00:19:55
triangles we had above were drawn in any
00:19:57
orientation that we wanted now we want
00:20:00
to start being more precise
00:20:02
and bring in more of the history of
00:20:03
trigonometry
00:20:05
we will start to draw all of our
00:20:06
triangles on the cartesian plane and
00:20:08
when we do this
00:20:09
we end up drawing in what's called
00:20:11
standard position
00:20:13
now here's what standard position is i'm
00:20:15
going to start off with just a regular
00:20:16
old angle okay here's an angle
00:20:19
and we often say that that's a
00:20:20
rotational angle so i start right here
00:20:24
and i rotate to here this first guy is
00:20:26
called the initial arm it's where i
00:20:28
start and then where i end up after the
00:20:30
rotation that's called my terminal arm
00:20:33
okay so to do something in standard
00:20:36
position
00:20:38
what we're going to do is we're going to
00:20:39
put our initial arm
00:20:42
on the positive x-axis so that's going
00:20:45
to look like that
00:20:47
the vertex goes at the origin
00:20:50
and then our terminal arm is going to be
00:20:52
wherever else wherever it ends up okay
00:20:54
could end up in any quadrant
00:20:57
and then we have to put the rotational
00:20:59
arrow in as well to say how we're
00:21:01
rotating so a positive rotation is
00:21:04
counterclockwise
00:21:06
a negative rotation would be in a
00:21:08
clockwise motion okay
00:21:12
so this is called angles in standard
00:21:14
position this guy here was just a
00:21:16
regular old angle it wasn't in standard
00:21:18
position
00:21:19
once i shift it so that the terminal arm
00:21:22
or sorry the initial arms on the
00:21:24
positive x-axis the vertex is at the
00:21:27
origin and the terminal arm is in one of
00:21:30
the other quadrants
00:21:32
now
00:21:32
it would be called an angle in standard
00:21:34
position
00:21:35
okay
00:21:39
okay so draw the following angles in
00:21:42
standard position on the cartesian plane
00:21:44
now when i ask you to do this i'm not
00:21:46
looking for perfection okay i'm looking
00:21:49
for just a rough sketch i want to know
00:21:50
you're in the right quadrant and for
00:21:53
instance 60 degrees if you did something
00:21:55
like this for 60 degrees i would say
00:21:58
yeah you're a little off right like that
00:22:00
very much is not 60 degrees but if you
00:22:03
did something like this
00:22:07
okay i would feel yeah you have a good
00:22:09
idea of where 60 degrees is
00:22:11
okay
00:22:12
so that's what we're looking for
00:22:14
um just as a clarification while we're
00:22:16
talking about this just in case you
00:22:18
don't remember the quadrants this would
00:22:19
be quadrant one quadrant two they are
00:22:22
often done in roman numerals okay
00:22:24
quadrant three
00:22:27
it's a lot of eyes to whoops sorry
00:22:29
that's a lot of ice to do and then
00:22:31
quadrant four is id
00:22:36
okay so it would be like that
00:22:40
so
00:22:41
here's my 60 degrees
00:22:45
200 well i know from here to here
00:22:48
is 180 so 200 will be just a little
00:22:50
farther than that
00:22:52
don't forget your rotational angle here
00:22:54
guys
00:22:55
um that's important right
00:22:59
110 i'd end up in quadrant two
00:23:03
and then negative 330 well you got to
00:23:05
watch with the negative that means i'm
00:23:07
going in a clockwise direction so i'm
00:23:08
going to be heading this way right so
00:23:09
this would be negative 90 i'm still
00:23:11
starting on the positive x so this would
00:23:13
be negative 90
00:23:15
negative 180
00:23:16
and then negative 270 so negative 330
00:23:20
actually puts me into quadrant 1 there
00:23:22
okay
00:23:26
perfect
00:23:28
now
00:23:29
state the measure of the acute angle
00:23:32
acute means less than 90 degrees
00:23:34
uh to the x-axis this is known as the
00:23:37
reference angle and the reference angle
00:23:39
is going to become very important to
00:23:41
your life over the next couple of days
00:23:43
okay
00:23:44
so let's look at quadrant one
00:23:47
uh when our terminal arm was in quadrant
00:23:49
one the reference angle and the angle
00:23:50
are actually the same thing okay so
00:23:53
there's no difference there
00:23:56
so it'd be 60 degrees in quadrant two
00:23:59
the quickest way back to the x-axis is
00:24:01
right here okay this angle here that i'm
00:24:04
drawing now i know from here to here is
00:24:07
180 and i know from here to here is 200
00:24:10
so that missing piece which is the
00:24:12
reference angle
00:24:13
um i would just go 200 minus 180
00:24:16
and get uh 20 there okay
00:24:22
um
00:24:23
very very important as we go through
00:24:24
this the reference angles are formed
00:24:26
with the x-axis
00:24:28
never the y-axis okay be very very
00:24:30
careful of that okay so for instance in
00:24:33
this third example don't look at this
00:24:35
and say oh there's my reference angle
00:24:36
nope it's not because you formed that
00:24:38
with the y-axis not with the x-axis
00:24:42
okay
00:24:43
so for this guy the reference angle is
00:24:45
actually right here
00:24:47
but again i know that this is 180 and
00:24:50
110 took me to the terminal arm so how
00:24:53
much do i have left to go to get to 180
00:24:55
that reference angle would have to be
00:24:56
70.
00:24:58
okay and then finally
00:25:00
here for the negative 330 i've almost
00:25:03
gone a full circle now just because my
00:25:05
angle is negative my reference angles
00:25:07
are never negative okay reference angles
00:25:09
are always considered positive
00:25:11
so i'm thinking to myself okay well
00:25:13
that's almost a full circle how much do
00:25:15
i have left to get to that full circle
00:25:17
well i'm missing that 30 degrees so 30
00:25:19
degrees there would be the reference
00:25:20
angle
00:25:21
okay
00:25:23
awesome
00:25:27
okay so
00:25:29
what would happen
00:25:30
uh to any of our angles above if we
00:25:32
rotated them another 360 or a negative
00:25:37
360 degrees
00:25:39
so here's the concept i want you to
00:25:41
think about let's just pick on this
00:25:42
angle here that i've drawn below for a
00:25:44
sec
00:25:45
um if i was to start at this terminal
00:25:48
arm
00:25:49
and rotate it another negative or
00:25:52
another positive 360 degrees sorry okay
00:25:57
the point i want to make is i end up
00:25:58
right where i started right because 360
00:26:01
degrees is a full circle
00:26:03
um so i'll end up right back there
00:26:06
that's the red line that i just drew
00:26:09
what if i started here
00:26:11
and i rotated
00:26:13
uh negative 360.
00:26:16
well that would end up right here
00:26:18
right right back where i started
00:26:21
so
00:26:23
what i want you to start thinking about
00:26:25
is for this one picture this one angle
00:26:28
and standard position that i've given
00:26:30
you there are actually multiple ways of
00:26:33
representing that angle okay the only
00:26:36
thing that would be different is the
00:26:38
specific rotational
00:26:41
angle that they show you there okay this
00:26:43
one is
00:26:45
uh just from here straight to here but i
00:26:48
could have started here
00:26:50
and gone and done a rotational angle
00:26:53
like that
00:26:54
okay
00:26:55
or i could have
00:26:57
started here
00:26:59
and done a rotational angle
00:27:02
like that
00:27:04
okay
00:27:05
i can do this multiple multiple times
00:27:08
and end up back where i started all of
00:27:10
those angles are called
00:27:12
coterminal to each other okay coterminal
00:27:15
whenever you see the word co
00:27:18
in front of a word in the english
00:27:20
language co means with so if you
00:27:22
cohabitate
00:27:24
uh you
00:27:25
live with someone
00:27:27
if you cooperate you work with someone
00:27:32
usually you work well with someone okay
00:27:35
um so all those angles are called
00:27:37
coterminal angles
00:27:38
now to get them i add 360 or i subtract
00:27:43
360 from the angle in question
00:27:46
[Music]
00:27:47
so name a positive angle and a negative
00:27:49
angle that are coterminal with 300
00:27:51
degrees
00:27:52
okay
00:27:53
uh then we'll write an equation to
00:27:55
represent all coterminal angles because
00:27:57
i'm hoping after that conversation we
00:27:59
just had that you're thinking hey isn't
00:28:01
there an infinite amount of ways to do
00:28:03
that and there is we'll talk about how
00:28:05
we write that in just one second let's
00:28:07
get the positive and the negative first
00:28:09
though so i want to start with the 300
00:28:12
degrees
00:28:13
so if i add 360 to that i'm right back
00:28:16
where i was for the 300. so 300 plus 360
00:28:21
that's going to give me 660.
00:28:23
okay or i could go 300 degrees minus
00:28:27
360. that would still give me a
00:28:29
coterminal angle of negative 60. okay so
00:28:32
in these examples i added one circle and
00:28:35
i subtracted one circle
00:28:37
but i could add five circles i could
00:28:39
subtract 20 circles i could add 18
00:28:42
billion circles okay
00:28:44
so
00:28:45
when we say we want to write an equation
00:28:48
to represent all coterminal angles this
00:28:51
is how we do it we start with the first
00:28:53
guy now that first guy by the way is
00:28:54
called a principal angle that's the
00:28:57
smallest positive angle we got okay
00:29:00
then we're going to say okay i want to
00:29:02
add multiples of a circle well a circle
00:29:05
is 360 degrees
00:29:07
and if i want any multiple of that i can
00:29:10
do that by putting an n there so 360
00:29:12
degrees n
00:29:14
okay
00:29:14
now
00:29:15
once i put a letter there i have to
00:29:18
establish the parameters around that
00:29:20
variable okay
00:29:22
for instance n can't be 1.2 if i went
00:29:26
1.2
00:29:28
circles
00:29:29
i'm not going to end up where i started
00:29:32
right it has to be a full circle so what
00:29:35
we do is we qualify that by saying okay
00:29:39
n has to be a member of the integer
00:29:41
family okay so it could be five it could
00:29:44
be ten it could be negative eighteen
00:29:45
billion but it can't be negative
00:29:47
eighteen billion point three
00:29:49
okay it has to be a full
00:29:52
circle
00:29:53
okay so that's gonna look like this
00:29:56
now you will see sometimes um
00:29:59
it said plus or minus here instead and
00:30:01
that's fine too okay often when it's
00:30:03
plus or minus they'll change the integer
00:30:06
family to the whole family because the
00:30:08
plus minus takes care of the negatives
00:30:12
so i don't need to say integers here um
00:30:14
it doesn't matter both are perfectly
00:30:16
acceptable
00:30:18
okay
00:30:20
okay so terminology i should now know or
00:30:24
soon know
00:30:25
well the initial arm um that is where we
00:30:29
start
00:30:30
um our rotational angle okay now if i'm
00:30:34
an angle in standard position
00:30:36
the initial arm would have to be on the
00:30:39
positive x-axis
00:30:41
okay so i'm just going to draw that
00:30:43
right here there's my initial arm
00:30:46
okay
00:30:47
and then the terminal arm would be
00:30:50
somewhere else
00:30:51
with the
00:30:53
vertex at the origin there okay now i
00:30:57
should i was using a straight line so
00:30:59
i'm just going to add arrowheads there
00:31:02
whoops that's a bad arrowhead but there
00:31:04
well that arrives that so i have an
00:31:06
angle in standard position um which is
00:31:08
the third one i need to talk about
00:31:10
anyway but don't forget you also have to
00:31:11
have the rotation
00:31:13
the angle of rotation on an angle of
00:31:16
standard position okay so we've checked
00:31:18
off initial arm we've checked off
00:31:20
terminal arm we've checked off standard
00:31:22
position now what i've just drawn you is
00:31:24
an angle in standard position okay
00:31:27
coterminal means they share the same
00:31:30
terminal arm okay so we could do the one
00:31:33
i just drew you but if i switched colors
00:31:36
we could also have represented it like
00:31:38
that and said it was the negative
00:31:41
version
00:31:42
or if i switched to another color i
00:31:44
could have said it was that spun around
00:31:47
a whole circle and then got there
00:31:49
okay that's what coterminal means
00:31:53
and then the primary trigonometric
00:31:54
ratios are
00:31:56
sine cosine and tangent and again it
00:31:58
might be very helpful to commit to
00:32:00
memory at this stage of your life the
00:32:02
sohcahtoa okay
00:32:04
that's just an acronym that means sine
00:32:07
is opposite over hypotenuse
00:32:10
cosine is adjacent over
00:32:12
hypotenuse and tangent is opposite over
00:32:16
adjacent
00:32:18
okay
00:32:19
so
00:32:20
that's our first lesson intrigue i've
00:32:23
written your homework there for those of
00:32:25
you who are with me at school
00:32:27
um and then we will get into some more
00:32:30
deeper level stuff um as we go through
00:32:33
the next couple of lessons the next
00:32:34
couple of lessons
00:32:36
kind of go
00:32:37
like
00:32:39
really closely together
00:32:41
so you know if you find one confusing
00:32:43
it's almost like you gotta get through
00:32:46
two to three of them before it kind of
00:32:48
all clicks um so don't stress if it
00:32:51
doesn't click right away
00:32:53
um just keep going through things keep
00:32:56
reviewing
00:32:57
um and give yourself a processing time
00:32:58
to click okay so uh get your homework
00:33:01
done be nice and neat communicate well
00:33:04
and check with me if you have any
00:33:05
questions okay take care guys bye