Find the exact value of the expression. $\csc \left( \cos ^ { - 1 } \frac { 24 } { 25 } \right)$
A) $28 ^ { \circ }$
B) $16 ^ { \circ }$
C) $- 1 ^ { \circ }$

D) $88 ^ { \circ }$
E)none of these

Question

Find the exact value of the expression. $\csc \left( \cos ^ { - 1 } \frac { 24 } { 25 } \right)$
A) $28 ^ { \circ }$
B) $16 ^ { \circ }$
C) $- 1 ^ { \circ }$

D) $88 ^ { \circ }$
E)none of these

Quizplus · Accepted Answer

The expression involves trigonometric functions and their inverses. To find the exact value of $\csc(\cos^{-1}(\frac{24}{25}))$, we first interpret $\cos^{-1}(\frac{24}{25})$ as the angle whose cosine is $\frac{24}{25}$. This angle is part of a right triangle where the adjacent side is 24, the hypotenuse is 25, and we can find the opposite side using the Pythagorean theorem: $opposite = \sqrt{25^2 - 24^2} = \sqrt{625 - 576} = \sqrt{49} = 7$. Thus, the sine of this angle is $\frac{opposite}{hypotenuse} = \frac{7}{25}$. Since $\csc$ is the reciprocal of $\sin$, $\csc(\cos^{-1}(\frac{24}{25})) = \frac{1}{\sin(\cos^{-1}(\frac{24}{25}))} = \frac{25}{7}$, which is not listed among the options.

Find the Exact Value of the Expression csc⁡(cos⁡−12425)\csc \left( \cos ^ { - 1 } \frac { 24 } { 25 } \right)csc(cos−12524​)

Find the Exact Value of the Expression $\csc \left( \cos ^ { - 1 } \frac { 24 } { 25 } \right)$