Textbook Question
In Exercises 35–60, find the reference angle for each angle. -13𝜋/3
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
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 guidance1
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).
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Inverse Tangent
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.
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5:04Converting between Degrees & Radians
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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4:45How to Use a Calculator for Trig Functions
Related Practice
Textbook Question
In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. 17𝜋 /5
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Textbook Question
In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. tan(-𝜋/4)
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Textbook Question
In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.
-210°
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Textbook Question
In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. -760°
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Textbook Question
In Exercises 63–68, find the exact value of each expression. Do not use a calculator. 1 + sin² 40° + sin² 50°
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