Use a calculator to find each value. Give answers as real numbers.
tan (arcsin 0.12251014)

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Recognize that the expression is \( \tan(\arcsin(0.12251014)) \). Here, \( \arcsin(0.12251014) \) represents an angle \( \theta \) whose sine is 0.12251014.

Set \( \theta = \arcsin(0.12251014) \), so by definition, \( \sin(\theta) = 0.12251014 \). Our goal is to find \( \tan(\theta) \).

Recall the Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \). Use this to find \( \cos(\theta) \) by calculating \( \cos(\theta) = \sqrt{1 - \sin^2(\theta)} \).

Once you have \( \cos(\theta) \), use the definition of tangent: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). Substitute the known values to express \( \tan(\theta) \).

Finally, use a calculator to evaluate the numerical value of \( \tan(\arcsin(0.12251014)) \) by performing the square root and division operations.

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Inverse Trigonometric Functions

Inverse trigonometric functions, like arcsin, return the angle whose trigonometric ratio matches a given value. For example, arcsin(0.12251014) gives the angle whose sine is 0.12251014, typically measured in radians or degrees.

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4:28

Introduction to Inverse Trig Functions

Relationship Between Sine and Tangent

The tangent of an angle can be expressed in terms of sine and cosine: tan(θ) = sin(θ)/cos(θ). Knowing the sine value allows calculation of cosine using the Pythagorean identity, enabling the evaluation of tangent for the given angle.

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Sine, Cosine, & Tangent of 30°, 45°, & 60°

Use of Calculators for Trigonometric Values

Calculators can compute inverse trigonometric functions and standard trigonometric functions accurately. Using a calculator, one can find arcsin values and then apply tangent to the resulting angle, ensuring answers are given as real numbers.

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