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Trigonometry Question

  1. Oct 2, 2015 #1
    For the triangle shown in the below image,

    ##\sin 60° = \frac{opposite}{hypotenuse} = \frac{y}{r}##
    ##\sin 30° = \frac{opposite}{hypotenuse} = \frac{x}{r}##

    The questions are:

    1. What is the opposite and hypotenuse of sin 90°?
    2. I am guessing that the opposite and hypotenuse of sin 90 is
    r and y respectively so that ##\sin 90° = \frac{opposite}{hypotenuse} = \frac{r}{y}##.
    Why sin 90° = 1?

    hypotenuse.png
     
  2. jcsd
  3. Oct 2, 2015 #2

    jedishrfu

    Staff: Mentor

    If you use the opposite over the hypotenuse definition then you'd get sin(90) = r/r = 1
     
  4. Oct 2, 2015 #3
    Do you mean that the sin 90° is: the opposite = r and the hypotenuse is = r too?
     
  5. Oct 2, 2015 #4

    jedishrfu

    Staff: Mentor

  6. Oct 2, 2015 #5
    How do you obtain ##\cos 90° = \frac{adjacent}{hypotenuse} = \frac{x}{r} = 0##?
     
  7. Oct 2, 2015 #6

    SteamKing

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    Look at your triangle. As the angle θ gets larger, the length of the base x gets smaller. The radius r remains constant. When θ = 90°, what must x be?
     
  8. Oct 2, 2015 #7
    There is no θ in my triangle. There are 90°, 60°, and 30° angles. Which one you meant as the θ?
     
  9. Oct 2, 2015 #8

    SteamKing

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    The angle which is labeled 60° in your diagram.

    The trig functions are defined as in this diagram:

    ttrig.gif
     
  10. Oct 3, 2015 #9
    Which side you meant by the radius?

    The θ you meant is the one labeled as 60°, but the cos 90° I meant is the right angle.

    What should the cos 90° be if the angle is the right angle, not the 60°?
     
  11. Oct 3, 2015 #10

    SteamKing

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    In your diagram of the triangle, the side labeled "r", which I indicated as "the radius r".

    Yes, but in order to calculate sine and cosine for 90°, you pretend to let the angle labeled 60° in your diagram increase to 90°, and examine what happens to the sides x and y in order for this to happen.

    cos (90°) = 0, always.

    Instead of using just a triangle to define the trig functions, often a unit circle (radius = 1) is used, like this:

    Trig_functions_on_unit_circle.PNG

    As the point P moves counterclockwise around the circle, the coordinates x and y of P are also the values of cos (θ) and sin (θ), respectively.

    Here is another diagram showing how sin (θ) varies for different angles θ drawn on a unit circle:

    3.png
     
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