Use the following formation to determine the remaining five trigonometric values. Rationalize any denominators that contain radicals. $\sec B = - \frac { 9 } { 4 } , 180 ^ { \circ }

Question

Use the following formation to determine the remaining five trigonometric values. Rationalize any denominators that contain radicals. $\sec B = - \frac { 9 } { 4 } , 180 ^ { \circ } < B < 270 ^ { \circ }$
A) $\sin B = - \frac { 4 } { 9 } , \tan B = \frac { 4 \sqrt { 65 } } { 65 } , \csc B = - \frac { 9 \sqrt { 65 } } { 65 } , \cot B = \frac { \sqrt { 65 } } { 4 } , \cos B = - \frac { \sqrt { 65 } } { 9 }$
B) $\sin B = - \frac { \sqrt { 65 } } { 9 } , \tan B = \frac { \sqrt { 65 } } { 4 } , \csc B = - \frac { 9 \sqrt { 65 } } { 65 } , \cot B = \frac { 4 \sqrt { 65 } } { 65 } , \cos B = - \frac { 4 } { 9 }$
C) $\sin B = \frac { \sqrt { 65 } } { 9 } , \tan B = - \frac { \sqrt { 65 } } { 4 } , \csc B = \frac { 9 \sqrt { 65 } } { 65 } , \cot B = - \frac { 4 \sqrt { 65 } } { 65 } , \cos B = \frac { 4 } { 9 }$
D) $\sin B = \frac { \sqrt { 65 } } { 9 } , \tan B = \frac { \sqrt { 65 } } { 4 } , \csc B = \frac { 9 \sqrt { 65 } } { 65 } , \cot B = \frac { 4 \sqrt { 65 } } { 65 } , \cos B = \frac { 4 } { 9 }$
E) $\sin B = - \frac { \sqrt { 65 } } { 9 } , \tan B = - \frac { \sqrt { 65 } } { 4 } , \csc \theta = - \frac { 9 \sqrt { 65 } } { 65 } , \cot B = - \frac { 4 \sqrt { 65 } } { 65 } , \cos B = - \frac { 4 } { 9 }$

Quizplus · Accepted Answer

Given $\sec B = -\frac{9}{4}$, we find $\cos B = -\frac{4}{9}$ since $\sec B = \frac{1}{\cos B}$. The angle $B$ is in the third quadrant where both sine and cosine are negative, which matches the signs of $\sin B$ and $\cos B$ in option B. Using Pythagoras' theorem, we find the hypotenuse as $9$ and the adjacent side as $4$, leading to the opposite side being $\sqrt{9^2 - 4^2} = \sqrt{65}$, making $\sin B = -\frac{\sqrt{65}}{9}$. The other trigonometric values are derived using these primary values, with correct signs for the third quadrant.

Use the Following Formation to Determine the Remaining Five Trigonometric sec⁡B=−94,180∘<B<270∘\sec B = - \frac { 9 } { 4 } , 180 ^ { \circ } < B < 270 ^ { \circ }secB=−49​,180∘<B<270∘

Use the Following Formation to Determine the Remaining Five Trigonometric $\sec B = - \frac { 9 } { 4 } , 180 ^ { \circ } < B < 270 ^ { \circ }$