Textbook Question
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
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 1, Problem 1.2.55
In Exercises 55–58, use a calculator to find the value of the acute angle θ to the nearest degree. sin θ = 0.2974
Verified step by step guidance1
Identify that the problem requires finding the acute angle \( \theta \) such that \( \sin \theta = 0.2974 \). Since \( \theta \) is acute, it lies between 0° and 90°.
Recall the inverse sine function (also called arcsine), which allows us to find the angle when the sine value is known: \( \theta = \sin^{-1}(0.2974) \).
Use a calculator set to degree mode to compute \( \sin^{-1}(0.2974) \). This will give the measure of \( \theta \) in degrees.
Round the result from the calculator to the nearest whole degree, as the problem requests the angle to the nearest degree.
Verify that the angle is indeed acute (between 0° and 90°) to confirm the solution is valid for the given problem.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Sine Function (Arcsin)
The inverse sine function, denoted as arcsin or sin⁻¹, is used to find the angle whose sine value is given. For a value y = sin θ, θ = arcsin(y) returns the angle θ in radians or degrees. This is essential for solving equations like sin θ = 0.2974 to find θ.
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Inverse Sine
Domain and Range of Sine and Arcsin
The sine function outputs values between -1 and 1, so the input to arcsin must lie within this range. The arcsin function returns angles only in the range of -90° to 90° (or -π/2 to π/2 radians), which corresponds to acute and some obtuse angles. Understanding this helps identify the correct angle solution.
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Using a Calculator to Find Angles
Calculators have a function to compute arcsin values, usually labeled as sin⁻¹. To find the angle θ from sin θ = 0.2974, input 0.2974 and use the inverse sine function. Ensure the calculator is set to the correct mode (degrees) to get the angle in degrees, as required.
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Related Practice
Textbook Question
In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. -150°
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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 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.
14𝜋/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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Textbook Question
In Exercises 71–74, find the length of the arc on a circle of radius r intercepted by a central angle θ. Express arc length in terms of 𝜋. Then round your answer to two decimal places. Radius, r: 8 feet Central Angle, θ: θ = 225°