In Exercises 55–58, use a calculator to find the value of the acute angle θ to the nearest degree. tan θ = 4.6252

Verified step by step guidance

1. Identify that the problem requires finding the acute angle \( \theta \) such that \( \tan \theta = 4.6252 \). Since \( \theta \) is acute, it lies between 0° and 90°.
2. Recall that to find an angle from its tangent value, you use the inverse tangent function (also called arctangent), denoted as \( \tan^{-1} \) or \( \arctan \).
3. Set up the equation \( \theta = \tan^{-1}(4.6252) \) to find the angle \( \theta \).
4. Use a calculator in degree mode to evaluate \( \tan^{-1}(4.6252) \). Make sure your calculator is set to degrees, not radians.
5. Round the resulting angle to the nearest whole degree to get the value of the acute angle \( \theta \).

Key Concepts

Tangent Function

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. It is a fundamental trigonometric function used to relate angles to side lengths. Understanding tan θ helps in finding the angle when the ratio is known.

Inverse Tangent (Arctan) Function

The inverse tangent function, denoted as arctan or tan⁻¹, is used to find the angle whose tangent is a given number. It allows us to determine the angle θ when tan θ is known, which is essential for solving the problem using a calculator.

Using a Calculator for Trigonometric Functions

Calculators can compute inverse trigonometric functions to find angles from ratios. It is important to ensure the calculator is set to the correct mode (degrees or radians) and to round the result appropriately, as the question asks for the angle to the nearest degree.