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Non-negative Acute Angle


Date: 01/26/99 at 20:47:21
From: Ben Stoerger
Subject: Algebra 2

Express as a function of a non-negative acute angle:

   sin(-260 degrees)

Evaluate each function leaving the result in radical form:

   cos(330 degrees)
   sin(-120 degrees)


Date: 01/27/99 at 12:10:12
From: Doctor Peterson
Subject: Re: Algebra 2

Hi, Ben. Let's see if I can give you a little study guide for this 
topic.

This is supposed to be a circle, showing the sine and cosine of angle 
A and several related angles.

                       *********
           180-A ******    |    ****** A
              ***----------+----------***
           ***  |\         |         /|  ***
          *     | \        |        / |     *
        **      |  \       |       /  |      **
       *        |   \      |      /   |        *
      *    sin A|    \     |     /    |sin A    *
     *          |     \    |    /     |          *
    *           |      \   |   /      |           *
    *           |       \  |  /       |           *
   *            |        \ | /        |            *
   *            |         \|/A        |            *
   *------------+----------*----------+------------*
   *            | -cos A  /|\  cos A  |            *
   *            |        / | \        |            *
    *           |       /  |  \       |           *
    *           |      /   |   \      |           *
     *          |     /    |    \     |          *
      *   -sin A|    /     |     \    |-sin A   *
       *        |   /      |      \   |        *
        **      |  /       |       \  |      **
          *     | /        |        \ |     *
           ***  |/         |         \|  ***
              ***----------+----------***
           180+A ******    |    ****** -A
                       *********

If we either add 360 degrees (a whole revolution) to A, or reflect A 
in the vertical axis by subtracting it from 180 degrees, we will still 
have the same sine.

If we reflect any of these angles in the horizontal axis by multiplying 
it by -1, we negate the sine.

Similar rules are true for the cosine.

You can see the same facts in terms of the graph of the sine. The sine 
function repeats with a period of 360 degrees, and also is symmetrical 
around 90, 270, ... degrees.

    |       ***                                 ***
    |    *   |   *                           *   |
    |  *     |     *                       *     |
    | *      |      *                     *      |
    |*       |       * 180+A       360-A *       |
    *--+-----+-----+--*--+-----+-----+--*-+------+--
    |  A         180-A *       |       * 360+A
    |                   *      |      *
    |                    *     |     *
    |                      *   |   *
    |                         ***
        I        II       III      IV        I    ...

Using these facts, we can move any angle into the first quadrant. 
(That's what a non-negative acute angle is.) 

For example, to find the sine of -240 (not quite your first problem) 
you can add 360 to get

    -240 + 360 = 120

which is in quadrant II, and then subtract this from 180 to get it 
into quadrant I:

    180 - 120 = 60

Neither of the things we did changes the sine, so the sine of -250 is 
the same as the sine of 60. If you were asked to express this in 
radical form, you would use the known sine of 60, sqrt(3)/2, as your 
answer.

In other cases, you will get the negative of the sine of a quadrant I 
angle; for instance, the sine of +240 is -sin(60). Can you see how to 
get that?

Let me know if you need more help. 

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Trigonometry

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