Determine the exact value of $\csc \theta$ when $\cos \theta = \frac { 5 } { 13 }$ and $\cot \theta

Question

Determine the exact value of $\csc \theta$ when $\cos \theta = \frac { 5 } { 13 }$ and $\cot \theta < 0$.
A) $\csc \theta = - \frac { 1 } { 12 }$
B) $\csc \theta = - \frac { 13 } { 12 }$
C) $\csc \theta = - \frac { 12 } { 13 }$
D) $\csc \theta = - \frac { 13 } { 6 }$
E) $\csc \theta = - \frac { 14 } { 11 }$

Quizplus · Accepted Answer

Given $\cos \theta = \frac{5}{13}$, we can find $\sin \theta$ using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$. Solving for $\sin \theta$ gives $\sin \theta = \pm \frac{12}{13}$. Since $\cot \theta < 0$ and $\cot \theta = \frac{\cos \theta}{\sin \theta}$, and given $\cos \theta > 0$, $\sin \theta$ must be negative for $\cot \theta$ to be negative. Thus, $\sin \theta = -\frac{12}{13}$. The cosecant function is the reciprocal of the sine function, so $\csc \theta = \frac{1}{\sin \theta} = -\frac{13}{12}$.

Determine the Exact Value Of csc⁡θ\csc \thetacscθ When cos⁡θ=513\cos \theta = \frac { 5 } { 13 }cosθ=135​

Determine the Exact Value Of $\csc \theta$ When $\cos \theta = \frac { 5 } { 13 }$