Point P is on the terminal side of θ . Evaluate the six trigonometric functions for θ. If the function is undefined, write "undefined."
-$\mathrm { P } ( - \sqrt { 3 } , - \sqrt { 7 } ) ;$ find $\csc \theta$
A) $= \frac { \sqrt { 30 } } { 3 }$
B) $\frac { \sqrt { 70 } } { 7 }$
C) $- \frac { \sqrt { 70 } } { 10 }$
D) $- \frac { \sqrt { 70 } } { 7 }$

Question

Quizplus · Accepted Answer

The cosecant function ($\csc \theta$) is the reciprocal of the sine function ($\sin \theta$), which is defined as the ratio of the length of the side opposite the angle ($y$) to the length of the hypotenuse ($r$). Given $P(-\sqrt{3}, -\sqrt{7})$, we find $r = \sqrt{(-\sqrt{3})^2 + (-\sqrt{7})^2} = \sqrt{3 + 7} = \sqrt{10}$. Thus, $\sin \theta = \frac{y}{r} = \frac{-\sqrt{7}}{\sqrt{10}}$, and $\csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{10}}{-\sqrt{7}} = -\frac{\sqrt{70}}{7}$.

Point P Is on the Terminal Side of θ P(−3,−7);\mathrm { P } ( - \sqrt { 3 } , - \sqrt { 7 } ) ;P(−3​,−7​);

Point P Is on the Terminal Side of θ $\mathrm { P } ( - \sqrt { 3 } , - \sqrt { 7 } ) ;$