Hello, qbkr21!

Find the value of $\displaystyle \cos\theta$ if $\displaystyle \sin\theta = \frac{8}{17}$ and $\displaystyle \cos\theta < 0$

Here is what I did.

$\displaystyle \cos = \frac{adj}{hyp}\quad\sin = \frac{opp}{hyp}$

Now I drew a right triangle and labeled the angle with respect to $\displaystyle \theta.$

So the hypotenese was 17 and $\displaystyle opp$ was 15.

Now to find the $\displaystyle adj$ side of the triangle I applied Pythagorus' theorem

and set it up like this: .$\displaystyle 8^2 + b^2\:=\:17^2$ solved for $\displaystyle b = 15$ **← ** here!

According to me, cosine should be $\displaystyle \frac{15}{17}$, but the computer won't accept the answer.

You overlooked one of the conditions: .$\displaystyle \cos\theta < 0$

. . If $\displaystyle \cos\theta$ is negative, then $\displaystyle \theta$ is in Quadrant 2 or 3.

Since $\displaystyle \sin\theta$ is positive, $\displaystyle \theta$ is in Quadrant 1 or 2.

. . Hence, $\displaystyle \theta$ is in Quadrant 2.

Your triangle should look like this: Code:

* |
| \ 17 |
8| \ |
| \ θ
- + - - - + - -
b |

Be very careful when using Pythagorus . . .

You had: .$\displaystyle 8^2 + b^2\:=\:17^2\quad\Rightarrow\quad b^2\:=\:225$

. . Then: .$\displaystyle b = $**±**$\displaystyle 15$

And you must choose the correct sign, you see.