Question

use the fundamental identities to find sin θ if cos θ = 2/3 and θ is in quadrant iv.
a. 5/4
b. 3√7/7
c. -3/2
d. -√5/3

Explanation:

Step1: Recall Pythagorean identity

$\sin^{2}\theta+\cos^{2}\theta = 1$.

Step2: Substitute the value of $\cos\theta$

Given $\cos\theta=\frac{2}{3}$, then $\sin^{2}\theta=1 - \cos^{2}\theta=1-(\frac{2}{3})^{2}=1-\frac{4}{9}=\frac{5}{9}$.

Step3: Determine the sign of $\sin\theta$

Since $\theta$ is in quadrant IV, $\sin\theta<0$. So $\sin\theta=-\sqrt{\frac{5}{9}}=-\frac{\sqrt{5}}{3}$.

Answer:

D. $-\frac{\sqrt{5}}{3}$