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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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