Question

Suppose sin(theta) = -3/7 and theta is in quadrant 4. Use a trig identity to find the value of cos(theta).

Answer 1

Since theta is in quadrant 4, the cosine of theta is positive.

Now, to solve this problem, we can use the following trigonometric identity:

`\begin{gathered} \cos ^(2)\theta+\sin ^2\theta=1 \\ \\ \cos ^2\theta=1-\sin ^(2)\theta \\ \\ \cos \theta=\sqrt[]{1-\sin ^(2)\theta} \end{gathered}`

Then, using

`\sin \theta=-(3)/(7)`

we obtain:

`\begin{gathered} \cos \theta=\sqrt[]{1-\mleft(-(3)/(7)\mright)^2} \\ \\ \cos \theta=\sqrt[]{1-(9)/(49)} \\ \\ \cos \theta=\sqrt[]{(49)/(49)-(9)/(49)} \\ \\ \cos \theta=\sqrt[]{(40)/(49)} \\ \\ \cos \theta=\frac{2\sqrt[]{10}}{7} \end{gathered}`

Therefore, the answer is

`\cos \theta=\frac{2\sqrt[]{10}}{7}`