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

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

`\cos (\theta_1)=-(5)/(13)`