Find the exact value of the indicated trigonometric function of θ.
-$\cos \theta = \frac { 2 } { 5 } , \quad \tan \theta

Question

Find the exact value of the indicated trigonometric function of θ.
-$\cos \theta = \frac { 2 } { 5 } , \quad \tan \theta < 0$
Find $\sin \theta$
A) $- \frac { 3 } { 2 }$
B) $- \frac { \sqrt { 5 } } { 2 }$
C) $- \sqrt { 5 }$
D) $- \frac { \sqrt { 21 } } { 5 }$

Quizplus · Accepted Answer

Given $\cos \theta = \frac{2}{5}$ and $\tan \theta < 0$, we can find $\sin \theta$ using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$. Substituting $\cos \theta = \frac{2}{5}$ gives $\sin^2 \theta = 1 - \left(\frac{2}{5}\right)^2 = 1 - \frac{4}{25} = \frac{21}{25}$. Taking the square root gives $\sin \theta = \pm \frac{\sqrt{21}}{5}$. Since $\tan \theta = \frac{\sin \theta}{\cos \theta}$ is negative, and $\cos \theta$ is positive, $\sin \theta$ must be negative. Therefore, $\sin \theta = -\frac{\sqrt{21}}{5}$.

Find the Exact Value of the Indicated Trigonometric Function of θ