Math 20-1 - Lesson 7.1 - intro to trigonometry
概要
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.
収穫
- 🔢 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.
タイムライン
- 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.
マインドマップ
ビデオ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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- 00:00:00hey friends
- 00:00:01so we're gonna start our new unit today
- 00:00:05uh which is on trick now uh for those of
- 00:00:08you who are with me at school
- 00:00:10um this is our last unit of 20-1
- 00:00:13if you are joining us from a different
- 00:00:16school this may not be your last unit uh
- 00:00:19but for us uh in my class this was how
- 00:00:22we are going to end our time together so
- 00:00:26pretty exciting uh you guys have come a
- 00:00:28long way and you've journeyed
- 00:00:30really well
- 00:00:31so pat yourself on the back
- 00:00:34um okay so
- 00:00:35the purpose of the first half of this
- 00:00:37lesson is just to kind of review some of
- 00:00:39the things we did in grade 10 in
- 00:00:42tennessee uh to make sure that you're
- 00:00:44okay with everything make sure we're all
- 00:00:46kind of going um starting off on the
- 00:00:48same page okay um the back half of this
- 00:00:51lesson we're gonna talk about angles in
- 00:00:53standard position and kind of introduce
- 00:00:54that concept but first um we just need
- 00:00:57to go back to sokatoa make sure
- 00:00:59everybody's uh cool with that um most of
- 00:01:01you no matter what school you would have
- 00:01:03gone to uh in grade 10 you would have
- 00:01:07had a formula sheet for math 10c that
- 00:01:09would have included the primary
- 00:01:11trigonometric ratios so sine
- 00:01:13is opposite over hypotenuse cosine is
- 00:01:16adjacent over hypotenuse and tangent is
- 00:01:18opposite over
- 00:01:20adjacent
- 00:01:21those aren't given to you anymore in
- 00:01:24grade 11. um so
- 00:01:26a very helpful acronym is sohcahtoa see
- 00:01:29how
- 00:01:30uh neat i can make my mouse writing be
- 00:01:34today sokatoa now this is just a quick
- 00:01:36little acronym to help you remember
- 00:01:39um
- 00:01:40who goes
- 00:01:41ah that was really bad oh okay who goes
- 00:01:44with what okay so if i just start at the
- 00:01:46left here and read with me the s is for
- 00:01:49sign and then the first uh
- 00:01:52or the next letter would be the one on
- 00:01:54top of the fraction for the ratio and
- 00:01:56then the third letter would be the one
- 00:01:57on the bottom of the fraction for the
- 00:01:59ratio so sine is opposite over
- 00:02:01hypotenuse
- 00:02:02cosine is adjacent over hypotenuse and
- 00:02:05then tangent is opposite over adjacent
- 00:02:08okay so sohcahtoa just a quick way to
- 00:02:11help you remember
- 00:02:12okay
- 00:02:13so
- 00:02:14um
- 00:02:16let's give it a quick go here we're
- 00:02:18gonna start with a very brief history
- 00:02:20and won't take too long on this but i
- 00:02:22think it's important that you know where
- 00:02:24some of the stuff comes from okay
- 00:02:26let's begin with a history of
- 00:02:27trigonometry which doesn't begin
- 00:02:29actually with triangles it actually
- 00:02:31begins with circles the study of
- 00:02:32trigonometry dates back to ancient
- 00:02:34babylon we're talking around 2000 bc
- 00:02:38as scholars attempted to understand the
- 00:02:39motion of the stars across the night sky
- 00:02:42really a lot of trigonometry was birthed
- 00:02:45from them trying to chart these stars
- 00:02:47and trying to say okay if the stars are
- 00:02:50in this position where am i sitting on a
- 00:02:52boat in the ocean or where am i sitting
- 00:02:54on this land mass and that's how we
- 00:02:56actually started mapping out the world
- 00:02:59they conjectured that the stars lie on a
- 00:03:02great sphere about the earth
- 00:03:05and consequently assume that their
- 00:03:07motions across the sky were along great
- 00:03:09circular arcs
- 00:03:11a natural question might be where on
- 00:03:14this circle
- 00:03:15must the star be if i'm viewing it at a
- 00:03:17particular angle of x so you can kind of
- 00:03:19see a person there staring up at the
- 00:03:21star um and thinking about what angle
- 00:03:24that forms
- 00:03:25okay
- 00:03:26uh and that's this the uh study of
- 00:03:30circle geometry was born
- 00:03:33trigonometry is actually one of the most
- 00:03:36used mathematical concepts um ever okay
- 00:03:40um
- 00:03:42it's used a lot in trades it's used in
- 00:03:45programming it's used in
- 00:03:48triangulation of
- 00:03:50cell phone data like the very fact that
- 00:03:54your cell phone can pinpoint where you
- 00:03:58are in the world up to about a meter
- 00:04:01um it's because your cell phone connects
- 00:04:03to three different towers okay
- 00:04:06and then how um
- 00:04:09how it connects to those three tower
- 00:04:11towers how it forms that triangle
- 00:04:13actually
- 00:04:14allows satellites to figure out exactly
- 00:04:17where that cell phone is
- 00:04:19in the vast world that's pretty
- 00:04:22incredible when you stop and think about
- 00:04:23it
- 00:04:24okay so the first thing we want to do is
- 00:04:26just review how to label triangles so
- 00:04:28i've given you a right triangle here and
- 00:04:30you know it's right because of the
- 00:04:32box down here by a
- 00:04:35we label angles as capital letters and
- 00:04:38we put those capital letters just on the
- 00:04:40outside edge so inside here is angle a
- 00:04:43and so we put a capital a on the corner
- 00:04:46inside here is angle b so we have a
- 00:04:48capital b on the corner and inside here
- 00:04:51is angle c and we have a capital c
- 00:04:54on the corner okay we also have this
- 00:04:56symbol here this symbol is called theta
- 00:04:59um and it's just used often in math for
- 00:05:02an unknown angle okay
- 00:05:04um
- 00:05:05so theta often used for an angle we
- 00:05:07don't know the measure
- 00:05:11now
- 00:05:12what we want to do is we want to put
- 00:05:14some words on here and we want to label
- 00:05:16these guys with lower case letters okay
- 00:05:19and the way we do lower case letters is
- 00:05:21we do the same letters as the
- 00:05:25angles so
- 00:05:26what i want you to do is i want you to
- 00:05:27put your pencil on a
- 00:05:29and i want you to just go away from it
- 00:05:32okay
- 00:05:34that will point to little a
- 00:05:37okay when we say go away we're talking
- 00:05:39actually about opposites okay so we say
- 00:05:42opposite angle a is little a
- 00:05:45opposite angle b well i put my pen on b
- 00:05:48and i go away from it
- 00:05:50it will point to the opposite side of b
- 00:05:52so that would be little b right here and
- 00:05:54then for c i put my angle at c and i go
- 00:05:57away from it well it's going to point to
- 00:06:00little c right here
- 00:06:02okay
- 00:06:03then we also have words for these guys
- 00:06:06um this one here you should know because
- 00:06:09he's opposite the 90 degree angle and
- 00:06:11you were introduced to that concept um a
- 00:06:14little while ago probably back in grade
- 00:06:168.
- 00:06:17um that's called the hypotenuse okay the
- 00:06:20hypotenuse in grade 8 you would have
- 00:06:24been introduced to the pythagorean
- 00:06:26theorem which says that the hypotenuse
- 00:06:28squared
- 00:06:29will equal the two legs squared and
- 00:06:32added together so the two other sides
- 00:06:34squared and added together okay
- 00:06:37now we also have words of opposite and
- 00:06:40adjacent that we need to talk about and
- 00:06:42they go depending on
- 00:06:45the non-ninety degree angle you're
- 00:06:47looking at so that's why i've stuck
- 00:06:48theta here
- 00:06:49okay
- 00:06:50um opposite theta is the opposite side
- 00:06:54okay
- 00:06:55and then
- 00:06:57so i've got the opposite i've got the
- 00:06:58hypotenuse over here i didn't actually
- 00:07:00label that sorry i talked about it but i
- 00:07:02didn't label it
- 00:07:03and then this guy here is adjacent
- 00:07:05adjacent means to
- 00:07:07be the side
- 00:07:09okay so the adjacent side is the one
- 00:07:12that helps form where that angle is but
- 00:07:15it's not the hypotenuse
- 00:07:17okay
- 00:07:18so now i'm going to clear that drawing
- 00:07:20and i'm just going to bring it a little
- 00:07:22neater
- 00:07:24okay here's everything i said there
- 00:07:27don't let me just go back for one sec
- 00:07:29sorry um
- 00:07:32previous okay and i'm just gonna hide my
- 00:07:35face here so you can see the word theta
- 00:07:37if you don't have the book at home
- 00:07:40okay
- 00:07:44awesome now
- 00:07:46um these are called the primary
- 00:07:48trigonometric ratios the sohcahtoa that
- 00:07:50i started you with
- 00:07:51the sine ratio the cosine ratio and the
- 00:07:53tangent ratio okay
- 00:07:55so the sine ratio is abbreviated sin on
- 00:07:59the on our calculator you'll see an sin
- 00:08:01button it's not a sin button we try not
- 00:08:04to sing
- 00:08:05it's a sine button so sine of theta is
- 00:08:07opposite over hypotenuse the cosine of
- 00:08:10theta is the ratio of the adjacent side
- 00:08:12over the hypotenuse side
- 00:08:14and then the tangent of theta is the
- 00:08:17ratio of the opposite divided by dg
- 00:08:22okay
- 00:08:24and the pythagorean theorem a squared is
- 00:08:26my hypotenuse for this guy be careful
- 00:08:28with the letters right the letters
- 00:08:29themselves don't matter
- 00:08:31what matters is whatever has been
- 00:08:32labeled as the hypotenuse that has to be
- 00:08:35by its side so it might be a in another
- 00:08:37situation it might be c then another
- 00:08:39situation might be w it doesn't really
- 00:08:41matter okay
- 00:08:42it's the hypotenuse squared
- 00:08:44equals the square of the other two legs
- 00:08:47the b and the c here
- 00:08:49um added together
- 00:08:51okay
- 00:08:52again i'm just gonna hide myself for a
- 00:08:54second so that you guys can see that at
- 00:08:55home
- 00:08:57um and then the other thing i need to
- 00:08:58remind you guys about is that you need
- 00:09:00to be in degree mode so there are
- 00:09:02two um
- 00:09:04ways we can measure
- 00:09:07just like if i showed you the messy desk
- 00:09:10that you can't see because it's right
- 00:09:11underneath the screen right now but i
- 00:09:13promise you it's messy there's piles of
- 00:09:15stuff everywhere um if i wanted to
- 00:09:18measure the length of my messy desk i
- 00:09:20have two options with which to measure
- 00:09:22it in i could measure it in centimeters
- 00:09:24or i could measure it in inches those
- 00:09:26would be two very different numbers but
- 00:09:27that doesn't change the length of my
- 00:09:29desk
- 00:09:30okay it's the same with angles when i go
- 00:09:33to measure an angle i have two different
- 00:09:35measurements i could use i could use or
- 00:09:38units of measure i should say i could
- 00:09:40use
- 00:09:41degrees or i could use radians now by
- 00:09:44default calculators graphing calculators
- 00:09:47are set in radians because radians is
- 00:09:49actually the much better mathematical
- 00:09:52um
- 00:09:53way to
- 00:09:54use the unit of measure for angles okay
- 00:09:58but we're not going to deal with that
- 00:10:00until grade 12. so for grade 11 we need
- 00:10:02to make sure your calculator is always
- 00:10:04in degree mode so that we can talk in
- 00:10:06degrees
- 00:10:07okay
- 00:10:08now to do that i've given some
- 00:10:10instructions here just to make sure
- 00:10:12everybody's good
- 00:10:14if you are a child of casio you'll go
- 00:10:17shift menu to get into your setup scroll
- 00:10:19down to angle and press f1 for degree
- 00:10:22if you are a child of ti
- 00:10:25you will press the mode button and
- 00:10:28go down three over one you'll notice if
- 00:10:30you go down three you have the words
- 00:10:31radian and degree and by default radian
- 00:10:34will be highlighted so then if you go
- 00:10:36over one
- 00:10:37um you'll be on the word degree and if
- 00:10:39you hit just hit enter it'll highlight
- 00:10:41degree instead
- 00:10:42as a check especially if you have one of
- 00:10:44these guys or if you have a scientific
- 00:10:47calculator and you want to make sure
- 00:10:48you're absolutely in the right mode 10
- 00:10:5245 should equal exactly one okay the tan
- 00:10:55of 45 should be one
- 00:10:58now uh kids get really uh stressed out
- 00:11:02about
- 00:11:03modes
- 00:11:04um
- 00:11:06so just for clarification
- 00:11:08the mode you're in doesn't matter unless
- 00:11:11you are pressing the sign the cosine or
- 00:11:13the tangent button okay
- 00:11:15no matter what mode you're in two plus
- 00:11:17two is going to equal four all the
- 00:11:19regular features of your calculator are
- 00:11:21going to work the only difference is
- 00:11:24when you go to press the sine the cosine
- 00:11:26or the tangent button okay
- 00:11:29all right now this phrase would have
- 00:11:31been introduced to you in uh grade 10 to
- 00:11:34solve a triangle means to determine the
- 00:11:37measure of the missing angles and the
- 00:11:39lengths of the missing sides
- 00:11:41in order to determine the unknown
- 00:11:43measures we use primary trigonometric
- 00:11:45ratios or the pythagorean theorem
- 00:11:47essentially
- 00:11:48um last year i could give you any right
- 00:11:51triangle
- 00:11:52there are six things is the first thing
- 00:11:54i need you to understand there right
- 00:11:55there's three angles and three sides so
- 00:11:57last year i could give you any right
- 00:11:58triangle um and i could give you three
- 00:12:01items so the 90 degree and maybe a side
- 00:12:04and an angle or the 90 degree and two
- 00:12:06sides or whatever
- 00:12:08and you could solve for the other three
- 00:12:10okay
- 00:12:11um this year we will actually be able to
- 00:12:14extend that to say it doesn't even have
- 00:12:16to be a 90 degree angle anymore there
- 00:12:18are other tools that we will learn in
- 00:12:20grade 11
- 00:12:21where i can give you any three items and
- 00:12:24you'll be able to get me the other three
- 00:12:26whether there's a 90 degree angle there
- 00:12:27or not
- 00:12:28okay
- 00:12:29all right so let's try a couple
- 00:12:32solve the following angle round side
- 00:12:34lengths to the nearest tenth and angles
- 00:12:36to the nearest degree it's super
- 00:12:38important that you pay attention to
- 00:12:40um
- 00:12:42rounding instructions okay
- 00:12:44side lengths have to be to the nearest
- 00:12:45tenth angles have to be to the nearest
- 00:12:47degree if you don't round this proper
- 00:12:49i'll take off half a point okay
- 00:12:51you have to follow the directions if you
- 00:12:53forget your units i'll take off half a
- 00:12:55point okay you have to talk about what
- 00:12:58unit of measure you're in so all of
- 00:13:00those things are super important okay
- 00:13:02so you have been given the 90 degree
- 00:13:04angle here you've been given little t
- 00:13:07here
- 00:13:07and you've been given
- 00:13:09a little w
- 00:13:11here
- 00:13:12so what you have to now do is find me
- 00:13:15angle f
- 00:13:16find me angle t
- 00:13:18and find me little f
- 00:13:20side f okay i'm gonna actually start
- 00:13:22with side f um i know
- 00:13:25two sides of a right triangle so i can
- 00:13:27get the third using the pythagorean
- 00:13:29theorem okay uh if i'm going to use the
- 00:13:32pythagorean theorem i need to think
- 00:13:34about the fact that i have the
- 00:13:35hypotenuse here so it's going to be f
- 00:13:37squared plus 7 squared equals 14 squared
- 00:13:41some of you from your junior high days
- 00:13:42may want to go directly to 14 squared
- 00:13:44minus 7 squared equals f squared that's
- 00:13:47totally fine as well okay
- 00:13:51so 14 squared equals seven and up seven
- 00:13:53squared plus f squared
- 00:13:56um so then
- 00:13:5714 squared is 196 7 squared is 49 i'll
- 00:14:00subtract 49 from both sides to get f
- 00:14:02squared and then the last step will be i
- 00:14:05need to uh square root it and follow the
- 00:14:08proper rounding construction so that's
- 00:14:10going to give me 12.1 centimeters okay
- 00:14:1312.1 centimeters for uh psi f
- 00:14:18now
- 00:14:19i could go ahead and solve for angle t
- 00:14:22or angle f
- 00:14:24if possible i'm not going to use this in
- 00:14:26my next step okay i could use it but
- 00:14:29what if i'm wrong
- 00:14:30if i'm wrong by default the next guy is
- 00:14:32going to be wrong right so try to only
- 00:14:34use what's been given to you
- 00:14:36uh whenever possible okay so if i wanted
- 00:14:40to get angle f
- 00:14:42okay the 7
- 00:14:44would be adjacent
- 00:14:47and
- 00:14:48i'm just going to put an a there
- 00:14:50and the 14
- 00:14:52would be
- 00:14:54the hypotenuse
- 00:14:56if i wanted to get angle
- 00:14:59t
- 00:15:02the 7
- 00:15:03would be my opposite
- 00:15:06yeah
- 00:15:07and the 14
- 00:15:09would be my hypotenuse okay so i'm going
- 00:15:12to set myself up
- 00:15:14to solve for f and to solve for angle t
- 00:15:17using
- 00:15:18cosine for angle f
- 00:15:20adjacent and hypotenuse that's what
- 00:15:22pairs adjacent and hypotenuse together
- 00:15:24is cosine
- 00:15:25and then sine for angle t because i need
- 00:15:28to pair up opposite end
- 00:15:30hypotenuse
- 00:15:31okay
- 00:15:34so i started with cos of f is 7 over 14.
- 00:15:38remember when i want to get f by itself
- 00:15:40i will arc cos or inverse cos this
- 00:15:43so f will be the inverse cose of one
- 00:15:46half i just reduced my seven over
- 00:15:47fourteen
- 00:15:48okay
- 00:15:50and then
- 00:15:51that gives me sixty degrees then for t
- 00:15:55i'm going to use sine
- 00:15:58sine of t is 7 over 14.
- 00:16:01so then i'm going to arc
- 00:16:03sine
- 00:16:04or
- 00:16:05inverse sine one half and get 30 degrees
- 00:16:09and of course as a check these two
- 00:16:12numbers added together should give me 90
- 00:16:14because i need 90 plus this 90 to get
- 00:16:17the 180 that all
- 00:16:19angles in a triangle have to add up to
- 00:16:21okay
- 00:16:23so
- 00:16:24i'm on a good path here
- 00:16:26um okay let me clear this ink and we'll
- 00:16:28do another example together
- 00:16:32so this time
- 00:16:34i have triangle drw
- 00:16:36where r angle r is 90 degrees
- 00:16:40little r is 12
- 00:16:42and the little w is 7. now the
- 00:16:44orientation doesn't matter okay
- 00:16:46you can orient orientate orient
- 00:16:49orientate you can orientate this
- 00:16:52triangle however you want okay
- 00:16:54um
- 00:16:56what matters is wherever you put that 90
- 00:16:59degrees you have to label that as r
- 00:17:02opposite that so the hypotenuse has to
- 00:17:04be labeled as 12.
- 00:17:06and then wherever you chose to write w
- 00:17:09opposite w has to be a seven
- 00:17:11okay
- 00:17:12so
- 00:17:13here's mine
- 00:17:15but again you have you may have your
- 00:17:17setup slightly different and that's
- 00:17:19totally fine
- 00:17:20okay
- 00:17:21so now
- 00:17:22i need to get
- 00:17:24a little d
- 00:17:29and i need to get angle w and i need to
- 00:17:31get angle d
- 00:17:33okay
- 00:17:34so a little d i'm going to use the
- 00:17:35pythagorean theorem again
- 00:17:41so i'll go 12 squared minus 7 squared
- 00:17:43essentially okay 12 squared's 144 7
- 00:17:47squared's 49 i'll subtract the 49 from
- 00:17:49both sides
- 00:17:52and then i'll square root it i got to
- 00:17:53follow rounding instructions this one
- 00:17:55said all answers should be to one
- 00:17:57decimal place so i've got 9.7
- 00:18:01okay
- 00:18:02now if i want angle w
- 00:18:06um i'm going to be looking at opposite
- 00:18:10and hypotenuse
- 00:18:13so that's going to use sine
- 00:18:16if i'm looking at angle d i'm going to
- 00:18:19look at adjacent
- 00:18:23and hypotenuse
- 00:18:25so that's going to be cosine okay
- 00:18:31so
- 00:18:32i started with
- 00:18:34angle d
- 00:18:35cosine of d is 7 over 12 i'm going to
- 00:18:38inverse cos or arc cos that's what gets
- 00:18:41rid of the coast that's attached to the
- 00:18:43d and allows d to be by itself
- 00:18:46so arc cos 7 over 12 and that's going to
- 00:18:49give me 54.3 again be careful with your
- 00:18:51rounding instructions it said all
- 00:18:53answers to one decimal place don't get
- 00:18:55caught into thinking that
- 00:18:57angles have to always be rounded to the
- 00:18:58nearest whole number you just do
- 00:19:00whatever the rounding instructions tell
- 00:19:01you to do
- 00:19:03okay
- 00:19:05and then for angle w
- 00:19:07we'll have sine of w is 7 over 12
- 00:19:11um and then we'll arc sine
- 00:19:147 over 12
- 00:19:16and get 35.7
- 00:19:19again as a check
- 00:19:22we know that
- 00:19:2435.7
- 00:19:26plus
- 00:19:27the
- 00:19:2854.3
- 00:19:29should give me 19. okay
- 00:19:32these two here should give me 90 and
- 00:19:34then when i add that to the other 90
- 00:19:36over here
- 00:19:38i'll get 180
- 00:19:40okay
- 00:19:43awesome
- 00:19:46okay
- 00:19:48so the triangles we had above i'm just
- 00:19:51going to kill my face here for sex so
- 00:19:53that you guys can read this with me the
- 00:19:55triangles we had above were drawn in any
- 00:19:57orientation that we wanted now we want
- 00:20:00to start being more precise
- 00:20:02and bring in more of the history of
- 00:20:03trigonometry
- 00:20:05we will start to draw all of our
- 00:20:06triangles on the cartesian plane and
- 00:20:08when we do this
- 00:20:09we end up drawing in what's called
- 00:20:11standard position
- 00:20:13now here's what standard position is i'm
- 00:20:15going to start off with just a regular
- 00:20:16old angle okay here's an angle
- 00:20:19and we often say that that's a
- 00:20:20rotational angle so i start right here
- 00:20:24and i rotate to here this first guy is
- 00:20:26called the initial arm it's where i
- 00:20:28start and then where i end up after the
- 00:20:30rotation that's called my terminal arm
- 00:20:33okay so to do something in standard
- 00:20:36position
- 00:20:38what we're going to do is we're going to
- 00:20:39put our initial arm
- 00:20:42on the positive x-axis so that's going
- 00:20:45to look like that
- 00:20:47the vertex goes at the origin
- 00:20:50and then our terminal arm is going to be
- 00:20:52wherever else wherever it ends up okay
- 00:20:54could end up in any quadrant
- 00:20:57and then we have to put the rotational
- 00:20:59arrow in as well to say how we're
- 00:21:01rotating so a positive rotation is
- 00:21:04counterclockwise
- 00:21:06a negative rotation would be in a
- 00:21:08clockwise motion okay
- 00:21:12so this is called angles in standard
- 00:21:14position this guy here was just a
- 00:21:16regular old angle it wasn't in standard
- 00:21:18position
- 00:21:19once i shift it so that the terminal arm
- 00:21:22or sorry the initial arms on the
- 00:21:24positive x-axis the vertex is at the
- 00:21:27origin and the terminal arm is in one of
- 00:21:30the other quadrants
- 00:21:32now
- 00:21:32it would be called an angle in standard
- 00:21:34position
- 00:21:35okay
- 00:21:39okay so draw the following angles in
- 00:21:42standard position on the cartesian plane
- 00:21:44now when i ask you to do this i'm not
- 00:21:46looking for perfection okay i'm looking
- 00:21:49for just a rough sketch i want to know
- 00:21:50you're in the right quadrant and for
- 00:21:53instance 60 degrees if you did something
- 00:21:55like this for 60 degrees i would say
- 00:21:58yeah you're a little off right like that
- 00:22:00very much is not 60 degrees but if you
- 00:22:03did something like this
- 00:22:07okay i would feel yeah you have a good
- 00:22:09idea of where 60 degrees is
- 00:22:11okay
- 00:22:12so that's what we're looking for
- 00:22:14um just as a clarification while we're
- 00:22:16talking about this just in case you
- 00:22:18don't remember the quadrants this would
- 00:22:19be quadrant one quadrant two they are
- 00:22:22often done in roman numerals okay
- 00:22:24quadrant three
- 00:22:27it's a lot of eyes to whoops sorry
- 00:22:29that's a lot of ice to do and then
- 00:22:31quadrant four is id
- 00:22:36okay so it would be like that
- 00:22:40so
- 00:22:41here's my 60 degrees
- 00:22:45200 well i know from here to here
- 00:22:48is 180 so 200 will be just a little
- 00:22:50farther than that
- 00:22:52don't forget your rotational angle here
- 00:22:54guys
- 00:22:55um that's important right
- 00:22:59110 i'd end up in quadrant two
- 00:23:03and then negative 330 well you got to
- 00:23:05watch with the negative that means i'm
- 00:23:07going in a clockwise direction so i'm
- 00:23:08going to be heading this way right so
- 00:23:09this would be negative 90 i'm still
- 00:23:11starting on the positive x so this would
- 00:23:13be negative 90
- 00:23:15negative 180
- 00:23:16and then negative 270 so negative 330
- 00:23:20actually puts me into quadrant 1 there
- 00:23:22okay
- 00:23:26perfect
- 00:23:28now
- 00:23:29state the measure of the acute angle
- 00:23:32acute means less than 90 degrees
- 00:23:34uh to the x-axis this is known as the
- 00:23:37reference angle and the reference angle
- 00:23:39is going to become very important to
- 00:23:41your life over the next couple of days
- 00:23:43okay
- 00:23:44so let's look at quadrant one
- 00:23:47uh when our terminal arm was in quadrant
- 00:23:49one the reference angle and the angle
- 00:23:50are actually the same thing okay so
- 00:23:53there's no difference there
- 00:23:56so it'd be 60 degrees in quadrant two
- 00:23:59the quickest way back to the x-axis is
- 00:24:01right here okay this angle here that i'm
- 00:24:04drawing now i know from here to here is
- 00:24:07180 and i know from here to here is 200
- 00:24:10so that missing piece which is the
- 00:24:12reference angle
- 00:24:13um i would just go 200 minus 180
- 00:24:16and get uh 20 there okay
- 00:24:22um
- 00:24:23very very important as we go through
- 00:24:24this the reference angles are formed
- 00:24:26with the x-axis
- 00:24:28never the y-axis okay be very very
- 00:24:30careful of that okay so for instance in
- 00:24:33this third example don't look at this
- 00:24:35and say oh there's my reference angle
- 00:24:36nope it's not because you formed that
- 00:24:38with the y-axis not with the x-axis
- 00:24:42okay
- 00:24:43so for this guy the reference angle is
- 00:24:45actually right here
- 00:24:47but again i know that this is 180 and
- 00:24:50110 took me to the terminal arm so how
- 00:24:53much do i have left to go to get to 180
- 00:24:55that reference angle would have to be
- 00:24:5670.
- 00:24:58okay and then finally
- 00:25:00here for the negative 330 i've almost
- 00:25:03gone a full circle now just because my
- 00:25:05angle is negative my reference angles
- 00:25:07are never negative okay reference angles
- 00:25:09are always considered positive
- 00:25:11so i'm thinking to myself okay well
- 00:25:13that's almost a full circle how much do
- 00:25:15i have left to get to that full circle
- 00:25:17well i'm missing that 30 degrees so 30
- 00:25:19degrees there would be the reference
- 00:25:20angle
- 00:25:21okay
- 00:25:23awesome
- 00:25:27okay so
- 00:25:29what would happen
- 00:25:30uh to any of our angles above if we
- 00:25:32rotated them another 360 or a negative
- 00:25:37360 degrees
- 00:25:39so here's the concept i want you to
- 00:25:41think about let's just pick on this
- 00:25:42angle here that i've drawn below for a
- 00:25:44sec
- 00:25:45um if i was to start at this terminal
- 00:25:48arm
- 00:25:49and rotate it another negative or
- 00:25:52another positive 360 degrees sorry okay
- 00:25:57the point i want to make is i end up
- 00:25:58right where i started right because 360
- 00:26:01degrees is a full circle
- 00:26:03um so i'll end up right back there
- 00:26:06that's the red line that i just drew
- 00:26:09what if i started here
- 00:26:11and i rotated
- 00:26:13uh negative 360.
- 00:26:16well that would end up right here
- 00:26:18right right back where i started
- 00:26:21so
- 00:26:23what i want you to start thinking about
- 00:26:25is for this one picture this one angle
- 00:26:28and standard position that i've given
- 00:26:30you there are actually multiple ways of
- 00:26:33representing that angle okay the only
- 00:26:36thing that would be different is the
- 00:26:38specific rotational
- 00:26:41angle that they show you there okay this
- 00:26:43one is
- 00:26:45uh just from here straight to here but i
- 00:26:48could have started here
- 00:26:50and gone and done a rotational angle
- 00:26:53like that
- 00:26:54okay
- 00:26:55or i could have
- 00:26:57started here
- 00:26:59and done a rotational angle
- 00:27:02like that
- 00:27:04okay
- 00:27:05i can do this multiple multiple times
- 00:27:08and end up back where i started all of
- 00:27:10those angles are called
- 00:27:12coterminal to each other okay coterminal
- 00:27:15whenever you see the word co
- 00:27:18in front of a word in the english
- 00:27:20language co means with so if you
- 00:27:22cohabitate
- 00:27:24uh you
- 00:27:25live with someone
- 00:27:27if you cooperate you work with someone
- 00:27:32usually you work well with someone okay
- 00:27:35um so all those angles are called
- 00:27:37coterminal angles
- 00:27:38now to get them i add 360 or i subtract
- 00:27:43360 from the angle in question
- 00:27:46[Music]
- 00:27:47so name a positive angle and a negative
- 00:27:49angle that are coterminal with 300
- 00:27:51degrees
- 00:27:52okay
- 00:27:53uh then we'll write an equation to
- 00:27:55represent all coterminal angles because
- 00:27:57i'm hoping after that conversation we
- 00:27:59just had that you're thinking hey isn't
- 00:28:01there an infinite amount of ways to do
- 00:28:03that and there is we'll talk about how
- 00:28:05we write that in just one second let's
- 00:28:07get the positive and the negative first
- 00:28:09though so i want to start with the 300
- 00:28:12degrees
- 00:28:13so if i add 360 to that i'm right back
- 00:28:16where i was for the 300. so 300 plus 360
- 00:28:21that's going to give me 660.
- 00:28:23okay or i could go 300 degrees minus
- 00:28:27360. that would still give me a
- 00:28:29coterminal angle of negative 60. okay so
- 00:28:32in these examples i added one circle and
- 00:28:35i subtracted one circle
- 00:28:37but i could add five circles i could
- 00:28:39subtract 20 circles i could add 18
- 00:28:42billion circles okay
- 00:28:44so
- 00:28:45when we say we want to write an equation
- 00:28:48to represent all coterminal angles this
- 00:28:51is how we do it we start with the first
- 00:28:53guy now that first guy by the way is
- 00:28:54called a principal angle that's the
- 00:28:57smallest positive angle we got okay
- 00:29:00then we're going to say okay i want to
- 00:29:02add multiples of a circle well a circle
- 00:29:05is 360 degrees
- 00:29:07and if i want any multiple of that i can
- 00:29:10do that by putting an n there so 360
- 00:29:12degrees n
- 00:29:14okay
- 00:29:14now
- 00:29:15once i put a letter there i have to
- 00:29:18establish the parameters around that
- 00:29:20variable okay
- 00:29:22for instance n can't be 1.2 if i went
- 00:29:261.2
- 00:29:28circles
- 00:29:29i'm not going to end up where i started
- 00:29:32right it has to be a full circle so what
- 00:29:35we do is we qualify that by saying okay
- 00:29:39n has to be a member of the integer
- 00:29:41family okay so it could be five it could
- 00:29:44be ten it could be negative eighteen
- 00:29:45billion but it can't be negative
- 00:29:47eighteen billion point three
- 00:29:49okay it has to be a full
- 00:29:52circle
- 00:29:53okay so that's gonna look like this
- 00:29:56now you will see sometimes um
- 00:29:59it said plus or minus here instead and
- 00:30:01that's fine too okay often when it's
- 00:30:03plus or minus they'll change the integer
- 00:30:06family to the whole family because the
- 00:30:08plus minus takes care of the negatives
- 00:30:12so i don't need to say integers here um
- 00:30:14it doesn't matter both are perfectly
- 00:30:16acceptable
- 00:30:18okay
- 00:30:20okay so terminology i should now know or
- 00:30:24soon know
- 00:30:25well the initial arm um that is where we
- 00:30:29start
- 00:30:30um our rotational angle okay now if i'm
- 00:30:34an angle in standard position
- 00:30:36the initial arm would have to be on the
- 00:30:39positive x-axis
- 00:30:41okay so i'm just going to draw that
- 00:30:43right here there's my initial arm
- 00:30:46okay
- 00:30:47and then the terminal arm would be
- 00:30:50somewhere else
- 00:30:51with the
- 00:30:53vertex at the origin there okay now i
- 00:30:57should i was using a straight line so
- 00:30:59i'm just going to add arrowheads there
- 00:31:02whoops that's a bad arrowhead but there
- 00:31:04well that arrives that so i have an
- 00:31:06angle in standard position um which is
- 00:31:08the third one i need to talk about
- 00:31:10anyway but don't forget you also have to
- 00:31:11have the rotation
- 00:31:13the angle of rotation on an angle of
- 00:31:16standard position okay so we've checked
- 00:31:18off initial arm we've checked off
- 00:31:20terminal arm we've checked off standard
- 00:31:22position now what i've just drawn you is
- 00:31:24an angle in standard position okay
- 00:31:27coterminal means they share the same
- 00:31:30terminal arm okay so we could do the one
- 00:31:33i just drew you but if i switched colors
- 00:31:36we could also have represented it like
- 00:31:38that and said it was the negative
- 00:31:41version
- 00:31:42or if i switched to another color i
- 00:31:44could have said it was that spun around
- 00:31:47a whole circle and then got there
- 00:31:49okay that's what coterminal means
- 00:31:53and then the primary trigonometric
- 00:31:54ratios are
- 00:31:56sine cosine and tangent and again it
- 00:31:58might be very helpful to commit to
- 00:32:00memory at this stage of your life the
- 00:32:02sohcahtoa okay
- 00:32:04that's just an acronym that means sine
- 00:32:07is opposite over hypotenuse
- 00:32:10cosine is adjacent over
- 00:32:12hypotenuse and tangent is opposite over
- 00:32:16adjacent
- 00:32:18okay
- 00:32:19so
- 00:32:20that's our first lesson intrigue i've
- 00:32:23written your homework there for those of
- 00:32:25you who are with me at school
- 00:32:27um and then we will get into some more
- 00:32:30deeper level stuff um as we go through
- 00:32:33the next couple of lessons the next
- 00:32:34couple of lessons
- 00:32:36kind of go
- 00:32:37like
- 00:32:39really closely together
- 00:32:41so you know if you find one confusing
- 00:32:43it's almost like you gotta get through
- 00:32:46two to three of them before it kind of
- 00:32:48all clicks um so don't stress if it
- 00:32:51doesn't click right away
- 00:32:53um just keep going through things keep
- 00:32:56reviewing
- 00:32:57um and give yourself a processing time
- 00:32:58to click okay so uh get your homework
- 00:33:01done be nice and neat communicate well
- 00:33:04and check with me if you have any
- 00:33:05questions okay take care guys bye
- trigonometry
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