Textbook Question
Which one of the following equations has solution π?
a. arccos (―1) = x
b. arccos 1 = x
c. arcsin (―1) = x
Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
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Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 5
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 5Chapter 7, Problem 5
Find the exact value of each real number y. Do not use a calculator.
y = sin⁻¹ √2/2
Verified step by step guidance1
Recognize that the problem asks for the exact value of \(y = \sin^{-1} \left( \frac{\sqrt{2}}{2} \right)\), which means we need to find the angle \(y\) whose sine is \(\frac{\sqrt{2}}{2}\).
Recall the range of the inverse sine function \(\sin^{-1} x\), which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This means the angle \(y\) must lie within this interval.
Identify the common angles where sine values are known exactly. The sine of \(\frac{\pi}{4}\) (or 45 degrees) is \(\frac{\sqrt{2}}{2}\).
Since \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\) and \(\frac{\pi}{4}\) lies within the range of \(\sin^{-1}\), conclude that \(y = \frac{\pi}{4}\).
Express the final answer as \(y = \frac{\pi}{4}\), which is the exact value of \(\sin^{-1} \left( \frac{\sqrt{2}}{2} \right)\) without using a calculator.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Sine Function (sin⁻¹ or arcsin)
The inverse sine function, sin⁻¹, returns the angle whose sine value is a given number. It is defined for inputs between -1 and 1 and outputs angles in the range [-π/2, π/2]. Understanding this helps find the angle y such that sin(y) = √2/2.
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Inverse Sine
Exact Values of Sine for Special Angles
Certain angles have well-known sine values expressed in exact radicals, such as sin(π/4) = √2/2. Recognizing these special angles allows you to determine the exact angle corresponding to a given sine value without a calculator.
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3:28Common Trig Functions For 45-45-90 Triangles
Domain and Range Restrictions of Inverse Trigonometric Functions
Inverse trig functions have restricted ranges to ensure they are functions. For sin⁻¹, the output angle y must lie between -π/2 and π/2. This restriction ensures a unique solution when finding y = sin⁻¹(√2/2).
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4:22Domain and Range of Function Transformations
Related Practice
Textbook Question
Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).
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sin x = ―√3/2
Textbook Question
The point (π/4, 1) lies on the graph of y = tan x. Therefore, the point _______ lies on the graph of y = tan⁻¹ x.
Textbook Question
Decide whether each statement is true or false. If false, explain why.
The tangent and secant functions are undefined for the same values.
Textbook Question
Solve each equation for all exact solutions, in degrees.
2√3 cos (θ/2) = -3
Textbook Question
Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.
√2 cos 2θ = -1