Question

Given that sin(theta)=15/17 and theta is in quadrant 2, determine sin(2theta), cos(2theta), and tan(2theta) Which quadrant is 2theta in?

Answer 1

Question #29898

Given that $\sin(\theta)$ = 15/17 and $\theta$ is in quadrant 2, determine $\sin(2\theta)$, $\cos(2\theta)$, and $\tan(2\theta)$ Which quadrant is 2theta in?

Solution.

If $\theta$ is in quadrant 2, then $\cos(\theta) \leq 0$ and $\cos(\theta) = -\sqrt{1 - \sin^2(\theta)} = -\sqrt{1 - \frac{225}{289}} = -\frac{8}{17}$.

Since $\sin (2\theta) = 2\sin (\theta)\cos (\theta)$, then

$$\sin (2 \theta) = 2 \cdot \frac {15}{17} \cdot \left(- \frac {8}{17}\right) = - \frac {240}{289}.$$

Using the formula $\cos (2\theta) = 2\cos^2 (\theta) - 1$, we obtain

$$\cos (2 \theta) = 2 \frac {64}{289} - 1 = - \frac {161}{289}.$$

Then $\tan (2\theta) = \frac{\sin(2\theta)}{\cos(2\theta)} = \frac{240}{161}$.

Taking into account that $\cos (2\theta) < 0$ and $\sin (2\theta) < 0$, we conclude that $2\theta$ is in quadrant 3.

Answer. $\sin (2\theta) = -\frac{240}{289}, \cos (2\theta) = -\frac{161}{289}, \tan (2\theta) = \frac{240}{161}$.