If \theta = \arcsin\left(\frac{3}{5}\right), where 0 \leq \theta \leq \frac{\pi}{2}, what … — ACT

If $\theta = \arcsin\left(\frac{3}{5}\right)$ , where $0 \leq \theta \leq \frac{\pi}{2}$ , what is the value of $\tan(\theta)$ ?

\frac{3}{4}

Since

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

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

. In a right-angled triangle with acute angle

\theta

, the opposite side is 3 and the hypotenuse is 5. By the Pythagorean theorem, the adjacent side is

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

. The tangent of

\theta

is defined as the ratio of the opposite side to the adjacent side, which gives

\frac{3}{4}

Interpret the inverse trigonometric equation

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

By definition of inverse sine,

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

means

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

for

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

Determine the side lengths of the reference right triangle

\text{opposite} = 3, \quad \text{hypotenuse} = 5, \quad \text{adjacent} = \sqrt{5^2 - 3^2} = 4

Using the ratio definition

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

and the Pythagorean theorem

a^2 + b^2 = c^2

to solve for the adjacent side.

Evaluate

\tan(\theta)

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}

The tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side.

Evaluating composite trigonometric expressions involving inverse functions using right triangle geometry.

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