Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
-Find $\csc B$ when $a = 9$ and $b = 7$.
A) $\frac { \sqrt { 130 } } { 7 }$

B) $\frac { 9 \sqrt { 130 } } { 130 }$

C) $\frac { 7 \sqrt { 130 } } { 130 }$

D) $\frac { \sqrt { 130 } } { 9 }$

Question

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
-Find $\csc B$ when $a = 9$ and $b = 7$.
A) $\frac { \sqrt { 130 } } { 7 }$

B) $\frac { 9 \sqrt { 130 } } { 130 }$

C) $\frac { 7 \sqrt { 130 } } { 130 }$

D) $\frac { \sqrt { 130 } } { 9 }$

Quizplus · Accepted Answer

The csc of an angle is the reciprocal of the sine of that angle. In a right triangle, $\sin B = \frac{opposite}{hypotenuse}$. Given $a = 9$ and $b = 7$, we first find the hypotenuse $c$ using Pythagoras' theorem: $c = \sqrt{a^2 + b^2} = \sqrt{9^2 + 7^2} = \sqrt{81 + 49} = \sqrt{130}$. Thus, $\sin B = \frac{a}{c} = \frac{9}{\sqrt{130}}$, and $\csc B = \frac{1}{\sin B} = \frac{\sqrt{130}}{9}$. However, since $B$ is adjacent to $b$, and $c$ is the hypotenuse, the correct setup should be $\sin B = \frac{b}{c} = \frac{7}{\sqrt{130}}$, leading to $\csc B = \frac{1}{\sin B} = \frac{\sqrt{130}}{7}$, which matches choice A.

Two Sides of a Right Triangle ABC (C Is the Right