Textbook Question
Solve each equation for exact solutions.
arcsin 2x + arccos x = π/6
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 6.49
Textbook Question
Use a calculator to approximate each value in decimal degrees.
θ = sin⁻¹ (-0.13349122)
Verified step by step guidance1
Understand that \( \sin^{-1} \) is the inverse sine function, also known as arcsin, which gives the angle whose sine is the given number.
Recognize that the value \(-0.13349122\) is within the range of the sine function, which is \([-1, 1]\).
Use a calculator to find \( \theta = \sin^{-1}(-0.13349122) \). Ensure the calculator is set to degree mode to get the angle in decimal degrees.
The calculator will provide an angle \( \theta \) in the range \([-90^\circ, 90^\circ]\) because the arcsin function returns values in this range.
Interpret the result as the angle whose sine is \(-0.13349122\), and ensure the angle is expressed in decimal degrees.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Sine Function
The inverse sine function, denoted as sin⁻¹ or arcsin, is used to find the angle whose sine is a given value. It is defined for inputs in the range of -1 to 1, producing outputs in the range of -90° to 90°. This function is essential for solving problems where the sine of an angle is known, and the angle itself needs to be determined.
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Calculator Functions
Using a scientific or graphing calculator effectively is crucial for approximating trigonometric values. Most calculators have a specific mode for trigonometric functions, and it is important to ensure that the calculator is set to the correct angle measurement (degrees or radians) before performing calculations. This ensures accurate results when using functions like sin⁻¹.
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Understanding Negative Values in Trigonometry
In trigonometry, a negative value for sine indicates that the angle is in the fourth quadrant when considering the unit circle. This is important for interpreting the results of the inverse sine function, as it helps to determine the correct angle that corresponds to the negative sine value, which will be between -90° and 0°.
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