If \theta = \arcsin\left(-\frac{3}{5}\right), what is the exact decimal value of \cos(2\th… — ACT

If $\theta = \arcsin\left(-\frac{3}{5}\right)$ , what is the exact decimal value of $\cos(2\theta)$ ?

Answer: 0.28

The exact decimal value of

\cos(2\theta)

0.28

Given

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

, the sine of

\theta

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

. Using the double-angle formula for cosine,

\cos(2\theta) = 1 - 2\sin^2(\theta)

, we substitute

\sin(\theta)

to get

\cos(2\theta) = 1 - 2\left(-\frac{3}{5}\right)^2 = 1 - \frac{18}{25} = \frac{7}{25} = 0.28

Identify the value of

\sin(\theta)

from the inverse trigonometric expression

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

By definition of the inverse sine function, if

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

, then

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

where

-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}

Select the double-angle identity for cosine that uses sine

\cos(2\theta) = 1 - 2\sin^2(\theta)

This form of the double-angle identity allows direct calculation without needing to calculate

\cos(\theta)

first.

Substitute

\sin(\theta)

and evaluate the expression

\cos(2\theta) = 0.28

Substituting

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

gives

1 - 2\left(-\frac{3}{5}\right)^2 = 1 - 2\left(\frac{9}{25}\right) = 1 - \frac{18}{25} = \frac{7}{25} = 0.28

Evaluating Trigonometric Functions of Inverse Trigonometric Expressions using Double-Angle Identities

Alternatively, place

\theta

in Quadrant IV (since

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

for a negative inverse sine input). The adjacent side is

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

, so

\cos(\theta) = \frac{4}{5}

. Then apply the alternative double-angle identity

\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = \left(\frac{4}{5}\right)^2 - \left(-\frac{3}{5}\right)^2 = \frac{16}{25} - \frac{9}{25} = \frac{7}{25} = 0.28

Estimated Time:1m 15s

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