Use the following information to determine the remaining five trigonometric values. Rationalize any denominators that contain radicals. $\cos \theta = - \frac { 4 } { 7 } , 90 ^ { \circ }

Question

Use the following information to determine the remaining five trigonometric values. Rationalize any denominators that contain radicals. $\cos \theta = - \frac { 4 } { 7 } , 90 ^ { \circ } < \theta < 180 ^ { \circ }$
A) $\sin \theta = \frac { \sqrt { 33 } } { 7 } , \tan \theta = - \frac { \sqrt { 33 } } { 4 } , \csc \theta = \frac { 7 \sqrt { 33 } } { 33 } , \cot \theta = - \frac { 4 \sqrt { 33 } } { 33 } , \sec \theta = - \frac { 7 } { 4 }$
B) $\sin \theta = \frac { \sqrt { 33 } } { 7 } , \tan \theta = - \frac { \sqrt { 33 } } { 4 } , \csc \theta = \frac { 7 \sqrt { 33 } } { 33 } , \cot \theta = - \frac { 4 \sqrt { 33 } } { 33 } , \sec \theta = \frac { 7 } { 4 }$
C) $\sin \theta = \frac { \sqrt { 33 } } { 7 } , \tan \theta = - \frac { \sqrt { 33 } } { 4 } , \csc \theta = - \frac { 7 } { 4 } , \cot \theta = - \frac { 4 \sqrt { 33 } } { 33 } , \sec \theta = \frac { 7 \sqrt { 33 } } { 33 }$
D) $\sin \theta = \frac { \sqrt { 33 } } { 7 } , \tan \theta = \frac { 4 \sqrt { 33 } } { 33 } , \csc \theta = - \frac { 7 } { 4 } , \cot \theta = \frac { \sqrt { 33 } } { 4 } , \sec \theta = \frac { 7 \sqrt { 33 } } { 33 }$
E) $\sin \theta = \frac { \sqrt { 33 } } { 7 } , \tan \theta = \frac { 4 \sqrt { 33 } } { 33 } , \csc \theta = \frac { 7 \sqrt { 33 } } { 33 } , \cot \theta = \frac { \sqrt { 33 } } { 4 } , \sec \theta = - \frac { 7 } { 4 }$

Quizplus · Accepted Answer

Given $\cos \theta = -\frac{4}{7}$ and $90^\circ < \theta < 180^\circ$, $\theta$ is in the second quadrant where $\sin \theta > 0$ and $\cos \theta < 0$. Using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$, we find $\sin \theta = \frac{\sqrt{33}}{7}$. The tangent ($\tan \theta$) is negative in the second quadrant, so $\tan \theta = -\frac{\sqrt{33}}{4}$. The reciprocal identities give $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, and $\cot \theta = \frac{1}{\tan \theta}$, leading to the values in choice A.

Use the Following Information to Determine the Remaining Five Trigonometric cos⁡θ=−47,90∘<θ<180∘\cos \theta = - \frac { 4 } { 7 } , 90 ^ { \circ } < \theta < 180 ^ { \circ }cosθ=−74​,90∘<θ<180∘

Use the Following Information to Determine the Remaining Five Trigonometric $\cos \theta = - \frac { 4 } { 7 } , 90 ^ { \circ } < \theta < 180 ^ { \circ }$