Use the fundamental identities to find the value of the trigonometric function.
-Find $\tan \theta$ if $\sec \theta = \frac { \sqrt { 26 } } { 5 }$ and $\sin \theta

Question

Use the fundamental identities to find the value of the trigonometric function.
-Find $\tan \theta$ if $\sec \theta = \frac { \sqrt { 26 } } { 5 }$ and $\sin \theta < 0$
A) $- 5$
B) $- \sqrt { 26 }$
C) $- \frac { 1 } { 5 }$
D) 5

Quizplus · Accepted Answer

Given $\sec \theta = \frac{\sqrt{26}}{5}$, we know $\cos \theta = \frac{5}{\sqrt{26}}$. Since $\sin \theta < 0$ and $\sec \theta > 0$, $\theta$ is in the fourth quadrant where $\sin \theta = -\sqrt{1 - \cos^2 \theta}$. Calculating $\sin \theta = -\sqrt{1 - \left(\frac{5}{\sqrt{26}}\right)^2} = -\frac{1}{\sqrt{26}}$. Thus, $\tan \theta = \frac{\sin \theta}{\cos \theta} = -\frac{1}{5}$.

Use the Fundamental Identities to Find the Value of the Trigonometric