1. ## Right triangle, find exact values w/sin addition formula

Check out the attached pic with the triangles and their data. I started out finding the missing sides using Pythagorem Theorem. Then I used the Law of Cosines to find angles A and B. I need to find the exact value of sin(A+B), but doing this doesn't result in answers that can use an addition or subtraction formula to get a "nice" exact value from the unit circle.....I've been kicked around by this problem for the last 2 days!!! Am I going about this wrong? If so, could someone shed some light on this for me?

Thanks,

~CC

2. Hello, calvin_coolidge!

Find the exact value of $\sin(A+B)$
Code:
      A
*
| *
|   *  21           *
5 |     *             |  *     __
|       *         3 |     * √58
|         *         |        *
* - - - - - *       * - - - - - * B
4√26                  7

In the left triangle, the third side is: . $\sqrt{21^2-5^2} \:=\:4\sqrt{26}$

. . Hence: . $\begin{array}{ccc}\sin A &=& \dfrac{4\sqrt{26}}{21} \\ \\[-3mm] \cos A &=& \dfrac{5}{21} \end{array}$

In the right triangle, the hypotenuse is: . $\sqrt{3^2+7^2} \:=\:\sqrt{58}$

. . Hence: . $\begin{array}{ccc}\sin B &=& \dfrac{3}{\sqrt{58}} \\ \\[-3mm] \cos B &=& \dfrac{7}{\sqrt{58}} \end{array}$

Substitute into: . $\sin(A+B) \;=\;\sin A\cos B + \sin B\cos A$