Write the product as a sum or difference of trigonometric functions.
-Find $\cot \theta$ if $\csc \theta = \frac { \sqrt { 10 } } { 3 }$ and $\theta$ is in quadrant $\mathrm { I }$.
A) $\frac { \sqrt { 10 } } { 10 }$
B) 3
C) $\sqrt { 10 }$
D) $\frac { 1 } { 3 }$

Question

Quizplus · Accepted Answer

In quadrant I, all trigonometric functions are positive. Given $\csc \theta = \frac{\sqrt{10}}{3}$, we have $\sin \theta = \frac{3}{\sqrt{10}}$. Using $\cot \theta = \frac{\cos \theta}{\sin \theta}$ and $\cos^2 \theta = 1 - \sin^2 \theta$, we find $\cos \theta = \frac{1}{\sqrt{10}}$. Thus, $\cot \theta = \frac{\frac{1}{\sqrt{10}}}{\frac{3}{\sqrt{10}}} = \frac{1}{3}$.

Write the Product as a Sum or Difference of Trigonometric cot⁡θ\cot \thetacotθ

Write the Product as a Sum or Difference of Trigonometric $\cot \theta$