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

Question

Evaluate without using a calculator.
-$\csc \theta$, if $\cot \theta = - \frac { 7 } { 10 }$ and $\cos \theta < 0$
A) $\frac { \sqrt { 149 } } { 10 }$
B) $\frac { 7 \sqrt { 149 } } { 149 }$
C) $- \frac { 7 \sqrt { 149 } } { 149 }$
D) $- \frac { \sqrt { 149 } } { 7 }$

Quizplus · Accepted Answer

Given $\cot \theta = -\frac{7}{10}$, we can find $\csc \theta$ by first finding $\sin \theta$. Since $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$, and $\cot \theta = -\frac{7}{10}$, we can represent $\sin \theta$ and $\cos \theta$ as opposite and adjacent sides of a right triangle, respectively. Thus, if $\cos \theta = -7$ and $\sin \theta = 10$ (considering the Pythagorean theorem and the given ratio), the hypotenuse (representing $\csc \theta = \frac{1}{\sin \theta}$) can be found using the Pythagorean theorem: $\sqrt{(-7)^2 + 10^2} = \sqrt{49 + 100} = \sqrt{149}$. Since $\cos \theta < 0$ and $\cot \theta < 0$, $\theta$ is in the second quadrant where $\sin \theta > 0$, making $\csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{149}}{10}$.

Evaluate Without Using a Calculator csc⁡θ\csc \thetacscθ , If cot⁡θ=−710\cot \theta = - \frac { 7 } { 10 }cotθ=−107​

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