Use the fundamental identities to find the value of the trigonometric function.
-$\sec \beta$, if $\sin \beta = - \frac { \sqrt { 5 } } { 4 }$, with $\beta$ in quadrant III
A) $- \frac { \sqrt { 11 } } { 4 }$
B) $- 4$
C) $- \frac { 4 \sqrt { 11 } } { 11 }$
D) $\frac { 4 \sqrt { 11 } } { 11 }$

Question

Quizplus · Accepted Answer

In quadrant III, both sine and secant are negative. Given $\sin \beta = - \frac { \sqrt { 5 } } { 4 }$, we can find $\cos \beta$ using the Pythagorean identity $\sin^2 \beta + \cos^2 \beta = 1$. Solving for $\cos \beta$, we get $\cos \beta = - \sqrt{1 - \sin^2 \beta} = - \sqrt{1 - \left(- \frac { \sqrt { 5 } } { 4 }\right)^2} = - \sqrt{1 - \frac{5}{16}} = - \sqrt{\frac{11}{16}} = - \frac{\sqrt{11}}{4}$. Since $\sec \beta = \frac{1}{\cos \beta}$, $\sec \beta = - \frac{4}{\sqrt{11}}$, which simplifies to $- \frac { 4 \sqrt { 11 } } { 11 }$.

Use the Fundamental Identities to Find the Value of the Trigonometric