Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 1.2.61

Chapter 1, Problem 1.2.61

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

Identify that the problem requires finding the acute angle \( \theta \) such that \( \tan \theta = 0.4169 \). Since \( \theta \) is acute, it lies between 0 and \( \frac{\pi}{2} \) radians.

Recall that to find an angle from its tangent value, you use the inverse tangent function (also called arctangent), denoted as \( \tan^{-1} \) or \( \arctan \). This function returns the angle whose tangent is the given number.

Set up the equation \( \theta = \tan^{-1}(0.4169) \). This means you will use a calculator to find the angle \( \theta \) in radians.

Make sure your calculator is set to radian mode before calculating \( \tan^{-1}(0.4169) \), as the problem asks for the angle in radians.

Use the calculator to find the value of \( \theta \), then round the result to three decimal places to get the final answer.

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Key Concepts

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

Inverse Tangent Function (arctan)

The inverse tangent function, denoted as arctan or tan⁻¹, is used to find an angle when the tangent value is known. It returns the angle whose tangent equals the given number, typically in radians or degrees. For tan θ = 0.4169, θ = arctan(0.4169).

Radian Measure of Angles

Radians are a way to measure angles based on the radius of a circle, where 2π radians equal 360 degrees. Using radians is common in calculus and higher mathematics. The problem requires the angle θ to be expressed in radians, rounded to three decimal places.

Using a Calculator for Trigonometric Functions

Calculators can compute inverse trigonometric functions to find angles from given ratios. Ensure the calculator is set to radian mode when the answer is required in radians. After computing arctan(0.4169), round the result to three decimal places as instructed.

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