In Exercises 59–62, use a calculator to find the value of the acute angle θ in radians, rounded to three decimal places. tan θ = 0.4169

Verified step by step guidance

1. Identify the trigonometric function involved: \( \tan \theta = 0.4169 \).
2. To find \( \theta \), use the inverse tangent function: \( \theta = \tan^{-1}(0.4169) \).
3. Ensure your calculator is set to radians mode, as the problem requires the angle in radians.
4. Input \( 0.4169 \) into your calculator and apply the inverse tangent function to find \( \theta \).
5. Round the result to three decimal places to obtain the value of the acute angle \( \theta \) in radians.

Key Concepts

Tangent Function

The tangent function, denoted as tan(θ), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It is also expressed as tan(θ) = sin(θ)/cos(θ). Understanding this function is crucial for solving problems involving angles and their corresponding ratios.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan or tan⁻¹, are used to find the angle when the value of a trigonometric function is known. For example, if tan(θ) = 0.4169, then θ can be found using θ = arctan(0.4169). This concept is essential for determining angles from given ratios.

Radians and Degrees

Radians and degrees are two units for measuring angles. Radians are based on the radius of a circle, where one full rotation (360 degrees) is equivalent to 2π radians. In this problem, the angle θ needs to be expressed in radians, which is important for accurate calculations and interpretations in trigonometry.