What is the exact decimal value of \tan\left(\arcsin\left(\frac{12}{13}\right)\right)? — ACT

What is the exact decimal value of $\tan\left(\arcsin\left(\frac{12}{13}\right)\right)$ ?

Answer: 2.4

The exact decimal value of

\tan\left(\arcsin\left(\frac{12}{13}\right)\right)

2.4

Letting

\theta = \arcsin\left(\frac{12}{13}\right)

, we establish a right triangle in Quadrant I with opposite side length

12

and hypotenuse length

13

. The adjacent side length is calculated via the Pythagorean theorem as

\sqrt{13^2 - 12^2} = 5

. The tangent of this angle is the ratio of the opposite side to the adjacent side,

\frac{12}{5}

, which equals

2.4

Define the angle using the inverse trigonometric expression

Let

\theta = \arcsin\left(\frac{12}{13}\right)

, which implies

\sin(\theta) = \frac{12}{13}

in Quadrant I.

The inverse sine function returns an angle whose sine is the given ratio, restricted to the principal interval

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

Determine

\cos(\theta)

using the fundamental trigonometric identity

\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left(\frac{12}{13}\right)^2} = \frac{5}{13}

Since

\theta

is in Quadrant I,

\cos(\theta)

is positive.

Evaluate

\tan(\theta)

as the ratio of sine to cosine

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{12/13}{5/13} = \frac{12}{5} = 2.4

Dividing opposite by adjacent (or sine by cosine) yields the exact decimal value

2.4

Evaluating algebraic values of composite inverse trigonometric expressions

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