Use the fundamental identities to find the value of the trigonometric function.
-$\csc \theta$, if $\cot \theta = - \frac { 7 } { 10 }$, with $\theta$ in quadrant II
A) $\frac { \sqrt { 149 } } { 10 }$
B) $- \frac { 7 \sqrt { 149 } } { 149 }$
C) $- \frac { \sqrt { 149 } } { 7 }$
D) $\frac { 7 \sqrt { 149 } } { 149 }$

Question

Quizplus · Accepted Answer

In quadrant II, $\sin \theta > 0$ and $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$. Given $\cot \theta = -\frac{7}{10}$, we can find $\sin \theta$ using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$. If $\cot \theta = -\frac{7}{10}$, then $\tan \theta = -\frac{10}{7}$, implying $\sin \theta = \frac{x}{\sqrt{x^2 + 7^2}}$ for some $x$. By substituting and solving, we find $\sin \theta = \frac{10}{\sqrt{149}}$, so $\csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{149}}{10}$.

Use the Fundamental Identities to Find the Value of the Trigonometric