What is the exact value of \cos\left(\arcsin\left(\frac{3}{5}\right)\right) expressed as a… — ACT

What is the exact value of $\cos\left(\arcsin\left(\frac{3}{5}\right)\right)$ expressed as a decimal?

Answer: 0.8

The exact value of the expression is 0.8.

Evaluating

\cos\left(\arcsin\left(\frac{3}{5}\right)\right)

requires finding the cosine of an angle

\theta

whose sine is

\frac{3}{5}

. In a right triangle with an opposite side of 3 and a hypotenuse of 5, the adjacent side is

\sqrt{5^2 - 3^2} = 4

. The cosine of

\theta

is the ratio of the adjacent side to the hypotenuse, which gives

\frac{4}{5} = 0.8

Interpret the inverse sine function as an angle.

Let

\theta = \arcsin\left(\frac{3}{5}\right)

, meaning

\sin(\theta) = \frac{3}{5}

for

0 < \theta < \frac{\pi}{2}

The inverse sine function returns an angle whose sine is the given value within the principal interval

\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]

Find the adjacent side of the right triangle associated with angle

\theta

\text{adjacent} = \sqrt{5^2 - 3^2} = \sqrt{16} = 4

By the Pythagorean theorem,

a^2 + b^2 = c^2

, so the adjacent side length is

\sqrt{c^2 - b^2}

Calculate the cosine of angle

\theta

\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} = 0.8

Cosine is defined as the ratio of the adjacent side to the hypotenuse in a right triangle.

Composition of Trigonometric and Inverse Trigonometric Functions

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