Textbook Question
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth of a degree, as appropriate.
sin 2θ = cos 2θ +1
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 7
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 7Chapter 7, Problem 7
Find the exact value of each real number y. Do not use a calculator.
y = tan⁻¹ (―√3)
Verified step by step guidance1
Recognize that the problem asks for the exact value of \(y = \tan^{-1}(-\sqrt{3})\), which means we need to find an angle \(y\) whose tangent is \(-\sqrt{3}\).
Recall the basic angles where tangent values are known: \(\tan(\frac{\pi}{3}) = \sqrt{3}\) and \(\tan(-\frac{\pi}{3}) = -\sqrt{3}\). These angles are commonly used in trigonometry and correspond to 60° and -60°, respectively.
Since the inverse tangent function \(\tan^{-1}(x)\) returns values in the interval \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) (or \((-90^\circ, 90^\circ)\)), the angle \(y\) must lie within this range.
Identify that \(y = -\frac{\pi}{3}\) is the angle in the principal range of \(\tan^{-1}\) such that \(\tan(y) = -\sqrt{3}\).
Therefore, the exact value of \(y\) is \(-\frac{\pi}{3}\), which corresponds to \(-60^\circ\) in degrees.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Tangent Function (arctan)
The inverse tangent function, denoted as tan⁻¹ or arctan, returns the angle whose tangent is a given number. It maps real numbers to angles typically in the range (-π/2, π/2). Understanding this helps find the angle y such that tan(y) equals the given value.
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Inverse Tangent
Exact Values of Tangent for Special Angles
Certain angles have well-known exact tangent values, such as π/6, π/4, and π/3. For example, tan(π/3) = √3 and tan(π/6) = 1/√3. Recognizing these values allows one to identify the angle corresponding to a given tangent without a calculator.
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Sign and Quadrant Considerations for arctan
Since tan⁻¹ returns angles in (-π/2, π/2), negative tangent values correspond to negative angles in this interval. For tan⁻¹(-√3), the angle is negative and matches the reference angle where tangent is √3, ensuring the correct sign and quadrant are chosen.
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Related Practice
Textbook Question
Solve each equation for x, where x is restricted to the given interval.
y = 3 tan 2x , for x in [―π/4, π/4]
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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cos θ = ―1/2
Textbook Question
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth of a degree, as appropriate.
3 cos² θ + 2 cos θ - 1 = 0
Textbook Question
Find the exact value of each real number y. Do not use a calculator.
y = cos⁻¹ (―√2/2)
Textbook Question
Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate.
tan² 2x -1 = 0