Textbook Question
In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. sin(-240°)
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 1, Problem 1.2.59
In Exercises 59–62, use a calculator to find the value of the acute angle θ in radians, rounded to three decimal places. cos θ = 0.4112
Verified step by step guidance1
Identify that the problem requires finding the acute angle \( \theta \) such that \( \cos \theta = 0.4112 \). Since \( \theta \) is acute, it lies between 0 and \( \frac{\pi}{2} \) radians.
Recall that to find \( \theta \) from \( \cos \theta = x \), you use the inverse cosine function: \( \theta = \cos^{-1}(x) \).
Set up the equation: \( \theta = \cos^{-1}(0.4112) \).
Use a calculator in radian mode to evaluate \( \cos^{-1}(0.4112) \). Make sure your calculator is set to radians, not degrees.
Round the result to three decimal places to get the value of the acute angle \( \theta \) in radians.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Cosine Function (arccos)
The inverse cosine function, denoted as arccos or cos⁻¹, is used to find the angle whose cosine value is given. It returns an angle in the range 0 to π radians (0° to 180°), which is essential for determining θ when cos θ is known.
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Inverse Cosine
Radian Measure of Angles
Radians are a way to measure angles based on the radius of a circle, where 2π radians equal 360 degrees. Understanding radians is crucial for interpreting and expressing the angle θ in the required unit, especially when using calculators set to radian mode.
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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. It is important to ensure the calculator is set to radian mode and to round the result correctly, here to three decimal places, to meet the problem’s requirements.
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4:45How to Use a Calculator for Trig Functions
Related Practice
Textbook Question
In Exercises 55–58, use a calculator to find the value of the acute angle θ to the nearest degree. sin θ = 0.2974
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Textbook Question
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. tan 120°
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Textbook Question
In Exercises 55–58, use a calculator to find the value of the acute angle θ to the nearest degree. tan θ = 4.6252
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Textbook Question
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. csc(-2𝜋/3)
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Textbook Question
If θ is an acute angle and cos θ = 1/3, find csc (𝜋/2 - θ).
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