What is the exact value of \cos\left(2\arctan(-3)\right) expressed as a decimal? — ACT

What is the exact value of $\cos\left(2\arctan(-3)\right)$ expressed as a decimal?

Answer: -0.8

The exact decimal value of

\cos\left(2\arctan(-3)\right)

-0.8

Letting

\theta = \arctan(-3)

gives

\tan(\theta) = -3

. Utilizing the double-angle formula

\cos(2\theta) = \frac{1 - \tan^2(\theta)}{1 + \tan^2(\theta)}

, we substitute

\tan(\theta) = -3

to obtain

\frac{1 - 9}{1 + 9} = \frac{-8}{10} = -0.8

Define an angle variable for the inverse trigonometric expression.

Let

\theta = \arctan(-3)

, which means

\tan(\theta) = -3

in Quadrant IV where

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

Applying the standard definition and principal range of the arctangent function.

Apply the double-angle identity for cosine expressed in terms of tangent.

\cos(2\theta) = \frac{1 - \tan^2(\theta)}{1 + \tan^2(\theta)}

This identity directly connects

\cos(2\theta)

to the known value of

\tan(\theta)

without needing radicals.

Substitute

\tan(\theta) = -3

into the identity and evaluate.

\cos(2\theta) = \frac{1 - 9}{1 + 9} = \frac{-8}{10} = -0.8

Simplifying the numerical expression produces the exact decimal value.

Inverse Trigonometric Functions and Double-Angle Identities

Estimated Time:1m 30s

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