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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

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

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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