Textbook Question
Find the exact value of each real number y. Do not use a calculator.
y = sin⁻¹ √2/2
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 6.3.23
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 6.3.23Chapter 7, Problem 6.3.23
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
Verified step by step guidance1
Start with the given equation: \(\sqrt{2} \cos 2\theta = -1\).
Isolate the cosine term by dividing both sides by \(\sqrt{2}\): \(\cos 2\theta = \frac{-1}{\sqrt{2}}\).
Recognize that \(\cos 2\theta = -\frac{1}{\sqrt{2}}\) corresponds to specific angles where the cosine value is negative and equal to \(-\frac{1}{\sqrt{2}}\).
Find the general solutions for \(2\theta\) by identifying the reference angle whose cosine is \(\frac{1}{\sqrt{2}}\), which is \(45^\circ\) or \(\frac{\pi}{4}\) radians, and then determine the angles in the appropriate quadrants where cosine is negative (quadrants II and III).
Divide the solutions for \(2\theta\) by 2 to solve for \(\theta\), and then restrict the solutions to the given interval \([0^\circ, 360^\circ)\) or \([0, 2\pi)\) depending on the unit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Double-Angle Trigonometric Functions
The equation involves cos 2θ, a double-angle function representing the cosine of twice the angle θ. Understanding how to manipulate and solve equations with double angles is essential, as it changes the period and affects the solution set within the given interval.
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05:06
Double Angle Identities
Solving Trigonometric Equations
Solving trigonometric equations requires isolating the trigonometric function and finding all angles within the specified interval that satisfy the equation. This often involves using inverse trigonometric functions and considering the periodic nature of sine and cosine.
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4:34How to Solve Linear Trigonometric Equations
Interval and Exact Solutions in Radians and Degrees
The problem specifies solutions over [0, 2π) for x and [0°, 360°) for θ, requiring careful attention to the domain. Providing exact solutions means expressing answers in terms of π or degrees rather than decimal approximations, ensuring precision and completeness.
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5:04Converting between Degrees & Radians
Related Practice
Textbook Question
Which one of the following equations has solution π?
a. arccos (―1) = x
b. arccos 1 = x
c. arcsin (―1) = x
Textbook Question
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
sec² θ tan θ = 2 tan θ
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.
8 sec² x/2 = 4
Textbook Question
The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval [0, 2π). Express solutions to four decimal places.
x² + sin x - x³ - cos x = 0
Textbook Question
Solve each equation for all exact solutions, in degrees.
2√3 cos (θ/2) = -3