Use the fundamental identities to find the value of the trigonometric function.
-$\tan \alpha$, if $\sec \alpha = \frac { 9 } { 4 }$, with $\alpha$ in quadrant IV
A) $- \frac { \sqrt { 65 } } { 4 }$
B) $- \frac { 9 } { 4 }$
C) $- \sqrt { 65 }$
D) $- \frac { \sqrt { 65 } } { 9 }$

Question

Quizplus · Accepted Answer

Given $\sec \alpha = \frac{9}{4}$, we know $\sec \alpha = \frac{1}{\cos \alpha}$, so $\cos \alpha = \frac{4}{9}$. In quadrant IV, $\sin \alpha$ is negative, and we can find $\sin \alpha$ using the Pythagorean identity $\sin^2 \alpha + \cos^2 \alpha = 1$. Solving for $\sin \alpha$, we get $\sin \alpha = -\frac{\sqrt{65}}{9}$. Therefore, $\tan \alpha = \frac{\sin \alpha}{\cos \alpha} = -\frac{\sqrt{65}}{4}$.

Use the Fundamental Identities to Find the Value of the Trigonometric