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tan (arcsin 0.12251014)

Verified step by step guidance

1. Identify the inner function: \( \arcsin(0.12251014) \). This represents the angle whose sine is 0.12251014.
2. Calculate \( \theta = \arcsin(0.12251014) \). This angle \( \theta \) is in the range \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\).
3. Use the identity \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). Since \( \sin(\theta) = 0.12251014 \), find \( \cos(\theta) \) using \( \cos(\theta) = \sqrt{1 - \sin^2(\theta)} \).
4. Substitute \( \sin(\theta) = 0.12251014 \) into the equation for \( \cos(\theta) \) to find \( \cos(\theta) = \sqrt{1 - (0.12251014)^2} \).
5. Finally, calculate \( \tan(\theta) = \frac{0.12251014}{\cos(\theta)} \) using the values obtained for \( \sin(\theta) \) and \( \cos(\theta) \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, are used to find the angle whose sine is a given value. For example, arcsin(0.12251014) gives the angle θ such that sin(θ) = 0.12251014. Understanding how to interpret these functions is crucial for solving problems involving angles and their corresponding trigonometric ratios.

Tangent Function

The tangent function, denoted as tan(θ), is defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed as tan(θ) = sin(θ) / cos(θ). Knowing how to calculate the tangent of an angle is essential for evaluating expressions that involve trigonometric functions.

Calculator Functions

Using a calculator effectively requires understanding how to input trigonometric functions and their inverses. Most scientific calculators have specific modes for degrees and radians, and knowing how to switch between these modes is important for obtaining accurate results. Familiarity with the calculator's functions will help in efficiently solving trigonometric problems.