Evaluate without using a calculator.
-$\sec \beta$, if $\sin \beta = - \frac { \sqrt { 10 } } { 9 }$, with $\beta$ in quadrant III
A) $- \frac { 9 \sqrt { 10 } } { \sqrt { 10 } }$
B) $- \frac { \sqrt { 71 } } { 9 }$
C) $\frac { 9 \sqrt { 71 } } { 71 }$
D) $= \frac { 9 \sqrt { 71 } } { 71 }$

Question

Quizplus · Accepted Answer

In quadrant III, both sine and cosine are negative, and secant is the reciprocal of cosine. Given $\sin \beta = - \frac { \sqrt { 10 } } { 9 }$, we find the cosine using the Pythagorean identity $\sin^2 \beta + \cos^2 \beta = 1$. Solving for $\cos \beta$, we get $\cos \beta = -\sqrt{1 - \sin^2 \beta} = -\sqrt{1 - \left(- \frac { \sqrt { 10 } } { 9 }\right)^2} = -\sqrt{1 - \frac{10}{81}} = -\sqrt{\frac{71}{81}} = -\frac{\sqrt{71}}{9}$. Therefore, $\sec \beta = \frac{1}{\cos \beta} = -\frac{9}{\sqrt{71}}$, which simplifies to option D when rationalized.

Evaluate Without Using a Calculator sec⁡β\sec \betasecβ , If sin⁡β=−109\sin \beta = - \frac { \sqrt { 10 } } { 9 }sinβ=−910​​

Evaluate Without Using a Calculator $\sec \beta$ , If $\sin \beta = - \frac { \sqrt { 10 } } { 9 }$