Math 20-1 - Lesson 7.1 - intro to trigonometry

00:33:10
https://www.youtube.com/watch?v=fUkzfUFhYOo

摘要

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.

心得

  • 🔢 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.

时间轴

  • 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.

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思维导图

视频问答

  • 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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  • 00:00:00
    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
  • 00:00:30
    really well
  • 00:00:31
    so pat yourself on the back
  • 00:00:34
    um okay so
  • 00:00:35
    the purpose of the first half of this
  • 00:00:37
    lesson is just to kind of review some of
  • 00:00:39
    the things we did in grade 10 in
  • 00:00:42
    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
  • 00:00:48
    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
  • 00:02:26
    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
  • 00:02:52
    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
  • 00:03:02
    great sphere about the earth
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    and consequently assume that their
  • 00:03:07
    motions across the sky were along great
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    circular arcs
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    a natural question might be where on
  • 00:03:14
    this circle
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    must the star be if i'm viewing it at a
  • 00:03:17
    particular angle of x so you can kind of
  • 00:03:19
    see a person there staring up at the
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    star um and thinking about what angle
  • 00:03:24
    that forms
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    okay
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    uh and that's this the uh study of
  • 00:03:30
    circle geometry was born
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    trigonometry is actually one of the most
  • 00:03:36
    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
  • 00:03:50
    cell phone data like the very fact that
  • 00:03:54
    your cell phone can pinpoint where you
  • 00:03:58
    are in the world up to about a meter
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    um it's because your cell phone connects
  • 00:04:03
    to three different towers okay
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    and then how um
  • 00:04:09
    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
  • 00:04:17
    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
  • 00:04:26
    just review how to label triangles so
  • 00:04:28
    i've given you a right triangle here and
  • 00:04:30
    you know it's right because of the
  • 00:04:32
    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
  • 00:04:40
    outside edge so inside here is angle a
  • 00:04:43
    and so we put a capital a on the corner
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    inside here is angle b so we have a
  • 00:04:48
    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
  • 00:05:02
    an unknown angle okay
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    um
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    so theta often used for an angle we
  • 00:05:07
    don't know the measure
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    now
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    what we want to do is we want to put
  • 00:05:14
    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
  • 00:05:25
    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
  • 00:06:57
    so i've got the opposite i've got the
  • 00:06:58
    hypotenuse over here i didn't actually
  • 00:07:00
    label that sorry i talked about it but i
  • 00:07:02
    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
  • 00:07:53
    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
  • 00:08:01
    button it's not a sin button we try not
  • 00:08:04
    to sing
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    it's a sine button so sine of theta is
  • 00:08:07
    opposite over hypotenuse the cosine of
  • 00:08:10
    theta is the ratio of the adjacent side
  • 00:08:12
    over the hypotenuse side
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    and then the tangent of theta is the
  • 00:08:17
    ratio of the opposite divided by dg
  • 00:08:22
    okay
  • 00:08:24
    and the pythagorean theorem a squared is
  • 00:08:26
    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
  • 00:08:37
    situation it might be c then another
  • 00:08:39
    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
  • 00:08:58
    remind you guys about is that you need
  • 00:09:00
    to be in degree mode so there are
  • 00:09:02
    two um
  • 00:09:04
    ways we can measure
  • 00:09:07
    just like if i showed you the messy desk
  • 00:09:10
    that you can't see because it's right
  • 00:09:11
    underneath the screen right now but i
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    promise you it's messy there's piles of
  • 00:09:15
    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
  • 00:09:26
    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
  • 00:09:44
    default calculators graphing calculators
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    are set in radians because radians is
  • 00:09:49
    actually the much better mathematical
  • 00:09:52
    um
  • 00:09:53
    way to
  • 00:09:54
    use the unit of measure for angles okay
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    but we're not going to deal with that
  • 00:10:00
    until grade 12. so for grade 11 we need
  • 00:10:02
    to make sure your calculator is always
  • 00:10:04
    in degree mode so that we can talk in
  • 00:10:06
    degrees
  • 00:10:07
    okay
  • 00:10:08
    now to do that i've given some
  • 00:10:10
    instructions here just to make sure
  • 00:10:12
    everybody's good
  • 00:10:14
    if you are a child of casio you'll go
  • 00:10:17
    shift menu to get into your setup scroll
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    down to angle and press f1 for degree
  • 00:10:22
    if you are a child of ti
  • 00:10:25
    you will press the mode button and
  • 00:10:28
    go down three over one you'll notice if
  • 00:10:30
    you go down three you have the words
  • 00:10:31
    radian and degree and by default radian
  • 00:10:34
    will be highlighted so then if you go
  • 00:10:36
    over one
  • 00:10:37
    um you'll be on the word degree and if
  • 00:10:39
    you hit just hit enter it'll highlight
  • 00:10:41
    degree instead
  • 00:10:42
    as a check especially if you have one of
  • 00:10:44
    these guys or if you have a scientific
  • 00:10:47
    calculator and you want to make sure
  • 00:10:48
    you're absolutely in the right mode 10
  • 00:10:52
    45 should equal exactly one okay the tan
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    of 45 should be one
  • 00:10:58
    now uh kids get really uh stressed out
  • 00:11:02
    about
  • 00:11:03
    modes
  • 00:11:04
    um
  • 00:11:06
    so just for clarification
  • 00:11:08
    the mode you're in doesn't matter unless
  • 00:11:11
    you are pressing the sign the cosine or
  • 00:11:13
    the tangent button okay
  • 00:11:15
    no matter what mode you're in two plus
  • 00:11:17
    two is going to equal four all the
  • 00:11:19
    regular features of your calculator are
  • 00:11:21
    going to work the only difference is
  • 00:11:24
    when you go to press the sine the cosine
  • 00:11:26
    or the tangent button okay
  • 00:11:29
    all right now this phrase would have
  • 00:11:31
    been introduced to you in uh grade 10 to
  • 00:11:34
    solve a triangle means to determine the
  • 00:11:37
    measure of the missing angles and the
  • 00:11:39
    lengths of the missing sides
  • 00:11:41
    in order to determine the unknown
  • 00:11:43
    measures we use primary trigonometric
  • 00:11:45
    ratios or the pythagorean theorem
  • 00:11:47
    essentially
  • 00:11:48
    um last year i could give you any right
  • 00:11:51
    triangle
  • 00:11:52
    there are six things is the first thing
  • 00:11:54
    i need you to understand there right
  • 00:11:55
    there's three angles and three sides so
  • 00:11:57
    last year i could give you any right
  • 00:11:58
    triangle um and i could give you three
  • 00:12:01
    items so the 90 degree and maybe a side
  • 00:12:04
    and an angle or the 90 degree and two
  • 00:12:06
    sides or whatever
  • 00:12:08
    and you could solve for the other three
  • 00:12:10
    okay
  • 00:12:11
    um this year we will actually be able to
  • 00:12:14
    extend that to say it doesn't even have
  • 00:12:16
    to be a 90 degree angle anymore there
  • 00:12:18
    are other tools that we will learn in
  • 00:12:20
    grade 11
  • 00:12:21
    where i can give you any three items and
  • 00:12:24
    you'll be able to get me the other three
  • 00:12:26
    whether there's a 90 degree angle there
  • 00:12:27
    or not
  • 00:12:28
    okay
  • 00:12:29
    all right so let's try a couple
  • 00:12:32
    solve the following angle round side
  • 00:12:34
    lengths to the nearest tenth and angles
  • 00:12:36
    to the nearest degree it's super
  • 00:12:38
    important that you pay attention to
  • 00:12:40
    um
  • 00:12:42
    rounding instructions okay
  • 00:12:44
    side lengths have to be to the nearest
  • 00:12:45
    tenth angles have to be to the nearest
  • 00:12:47
    degree if you don't round this proper
  • 00:12:49
    i'll take off half a point okay
  • 00:12:51
    you have to follow the directions if you
  • 00:12:53
    forget your units i'll take off half a
  • 00:12:55
    point okay you have to talk about what
  • 00:12:58
    unit of measure you're in so all of
  • 00:13:00
    those things are super important okay
  • 00:13:02
    so you have been given the 90 degree
  • 00:13:04
    angle here you've been given little t
  • 00:13:07
    here
  • 00:13:07
    and you've been given
  • 00:13:09
    a little w
  • 00:13:11
    here
  • 00:13:12
    so what you have to now do is find me
  • 00:13:15
    angle f
  • 00:13:16
    find me angle t
  • 00:13:18
    and find me little f
  • 00:13:20
    side f okay i'm gonna actually start
  • 00:13:22
    with side f um i know
  • 00:13:25
    two sides of a right triangle so i can
  • 00:13:27
    get the third using the pythagorean
  • 00:13:29
    theorem okay uh if i'm going to use the
  • 00:13:32
    pythagorean theorem i need to think
  • 00:13:34
    about the fact that i have the
  • 00:13:35
    hypotenuse here so it's going to be f
  • 00:13:37
    squared plus 7 squared equals 14 squared
  • 00:13:41
    some of you from your junior high days
  • 00:13:42
    may want to go directly to 14 squared
  • 00:13:44
    minus 7 squared equals f squared that's
  • 00:13:47
    totally fine as well okay
  • 00:13:51
    so 14 squared equals seven and up seven
  • 00:13:53
    squared plus f squared
  • 00:13:56
    um so then
  • 00:13:57
    14 squared is 196 7 squared is 49 i'll
  • 00:14:00
    subtract 49 from both sides to get f
  • 00:14:02
    squared and then the last step will be i
  • 00:14:05
    need to uh square root it and follow the
  • 00:14:08
    proper rounding construction so that's
  • 00:14:10
    going to give me 12.1 centimeters okay
  • 00:14:13
    12.1 centimeters for uh psi f
  • 00:14:18
    now
  • 00:14:19
    i could go ahead and solve for angle t
  • 00:14:22
    or angle f
  • 00:14:24
    if possible i'm not going to use this in
  • 00:14:26
    my next step okay i could use it but
  • 00:14:29
    what if i'm wrong
  • 00:14:30
    if i'm wrong by default the next guy is
  • 00:14:32
    going to be wrong right so try to only
  • 00:14:34
    use what's been given to you
  • 00:14:36
    uh whenever possible okay so if i wanted
  • 00:14:40
    to get angle f
  • 00:14:42
    okay the 7
  • 00:14:44
    would be adjacent
  • 00:14:47
    and
  • 00:14:48
    i'm just going to put an a there
  • 00:14:50
    and the 14
  • 00:14:52
    would be
  • 00:14:54
    the hypotenuse
  • 00:14:56
    if i wanted to get angle
  • 00:14:59
    t
  • 00:15:02
    the 7
  • 00:15:03
    would be my opposite
  • 00:15:06
    yeah
  • 00:15:07
    and the 14
  • 00:15:09
    would be my hypotenuse okay so i'm going
  • 00:15:12
    to set myself up
  • 00:15:14
    to solve for f and to solve for angle t
  • 00:15:17
    using
  • 00:15:18
    cosine for angle f
  • 00:15:20
    adjacent and hypotenuse that's what
  • 00:15:22
    pairs adjacent and hypotenuse together
  • 00:15:24
    is cosine
  • 00:15:25
    and then sine for angle t because i need
  • 00:15:28
    to pair up opposite end
  • 00:15:30
    hypotenuse
  • 00:15:31
    okay
  • 00:15:34
    so i started with cos of f is 7 over 14.
  • 00:15:38
    remember when i want to get f by itself
  • 00:15:40
    i will arc cos or inverse cos this
  • 00:15:43
    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
标签
  • trigonometry
  • SOHCAHTOA
  • angles
  • history
  • right triangles
  • standard position
  • Cartesian plane
  • coterminal angles
  • calculator settings
  • educational lesson