Question

You might need: CalculatorThe angle O, is located in Quadrant III, and sin((.)1213What is the value of cos((,)?Express your answer exactly.cos((.) =

Answer 1

We know that:

$\sin (\theta_1)=-\frac{12}{13}$

There is also an interesting property that relates the sine and the cosine of an angle:

$\sin ^2(\theta_1)+\cos ^2(\theta_1)=1$

We can find the cosine of theta using this equation:

$\begin{gathered} \cos ^2(\theta_1)=1-\sin ^2(\theta_1) \\ \cos (\theta_1)=\sqrt{1-\sin^2(\theta_1)} \\ \cos (\theta_1)=\sqrt[]{1-(-\frac{12}{13})^2} \\ \lvert\cos (\theta_1)\rvert=\sqrt[]{1-\frac{144}{169}}=\sqrt[]{\frac{25}{169}} \\ \lvert\cos (\theta_1)\rvert=\frac{5}{13} \end{gathered}$

Since theta is in the third quadrant then its cosine must be a negative number so:

$\cos (\theta_1)=-\frac{5}{13}$