Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Find the unknown side length
using the Pythagorean theorem and then find the value of the indicated trigonometric function of the given angle.
Rationalize the denominator if applicable.
-Find $\csc B$ when $a = 5$ and $b = 4$.
A) $\frac { \sqrt { 41 } } { 5 }$
B) $\frac { \sqrt { 41 } } { 4 }$
C) $\frac { 5 \sqrt { 41 } } { 41 }$
D) $\frac { 4 \sqrt { 41 } } { 41 }$

Question

Quizplus · Accepted Answer

First, use the Pythagorean theorem to find the length of side c: $c^2 = a^2 + b^2 = 5^2 + 4^2 = 25 + 16 = 41$, so $c = \sqrt{41}$. The cosecant function is defined as $\csc(\theta) = \frac{1}{\sin(\theta)}$, and since $\sin(B) = \frac{opposite}{hypotenuse} = \frac{a}{c} = \frac{5}{\sqrt{41}}$, then $\csc(B) = \frac{1}{\sin(B)} = \frac{\sqrt{41}}{5}$. However, the correct answer is $\csc(B) = \frac{1}{\sin(B)} = \frac{\sqrt{41}}{4}$ because $\sin(B) = \frac{opposite}{hypotenuse} = \frac{b}{c} = \frac{4}{\sqrt{41}}$, not $\frac{5}{\sqrt{41}}$. My initial explanation mistakenly used side a instead of b for the opposite side when calculating $\sin(B)$.

Suppose ABC Is a Right Triangle with Sides of Lengths csc⁡B\csc BcscB

Suppose ABC Is a Right Triangle with Sides of Lengths $\csc B$