Evaluate without using a calculator.
-$\cot \gamma$, if $\csc \gamma = - \frac { 10 } { 7 }$ and $\cos \gamma

Question

Evaluate without using a calculator.
-$\cot \gamma$, if $\csc \gamma = - \frac { 10 } { 7 }$ and $\cos \gamma < 0$
A) $\frac { \sqrt { 51 } } { 7 }$
B) $- \frac { \sqrt { 51 } } { 10 }$
C) $- \frac { 10 \sqrt { 51 } } { 51 }$
D) $- \frac { 7 \sqrt { 51 } } { 51 }$

Quizplus · Accepted Answer

Given $\csc \gamma = -\frac{10}{7}$, we find $\sin \gamma = -\frac{7}{10}$. To find $\cot \gamma = \frac{\cos \gamma}{\sin \gamma}$, we first find $\cos \gamma$ using $\sin^2 \gamma + \cos^2 \gamma = 1$. Substituting $\sin \gamma = -\frac{7}{10}$ gives $\cos \gamma = \pm \frac{\sqrt{51}}{10}$. Since $\cos \gamma < 0$, $\cos \gamma = -\frac{\sqrt{51}}{10}$. Thus, $\cot \gamma = \frac{\cos \gamma}{\sin \gamma} = \frac{-\frac{\sqrt{51}}{10}}{-\frac{7}{10}} = \frac{\sqrt{51}}{7}$.

Evaluate Without Using a Calculator cot⁡γ\cot \gammacotγ , If csc⁡γ=−107\csc \gamma = - \frac { 10 } { 7 }cscγ=−710​

Evaluate Without Using a Calculator $\cot \gamma$ , If $\csc \gamma = - \frac { 10 } { 7 }$