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]
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 11
Lial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 11Chapter 7, Problem 11
Find the exact value of each real number y. Do not use a calculator.
y = sec⁻¹ (―2)
Verified step by step guidance1
Recall that the function \( y = \sec^{-1}(x) \) is the inverse secant function, which gives an angle \( y \) such that \( \sec y = x \). Here, we want to find \( y \) such that \( \sec y = -2 \).
Use the identity relating secant and cosine: \( \sec y = \frac{1}{\cos y} \). So, \( \sec y = -2 \) implies \( \frac{1}{\cos y} = -2 \), which means \( \cos y = -\frac{1}{2} \).
Determine the range of \( y = \sec^{-1}(x) \). By definition, \( y \) lies in \( [0, \pi] \) excluding \( \frac{\pi}{2} \), because secant is not defined at \( \frac{\pi}{2} \).
Find all angles \( y \) in the interval \( [0, \pi] \) where \( \cos y = -\frac{1}{2} \). Recall the unit circle values where cosine equals \( -\frac{1}{2} \).
Select the angle(s) from the previous step that satisfy the domain of \( \sec^{-1} \) and write the exact value(s) of \( y \) accordingly.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Secant Function (sec⁻¹ or arcsec)
The inverse secant function, sec⁻¹(x), returns the angle whose secant is x. It is defined for |x| ≥ 1, and its range is typically [0, π] excluding π/2. Understanding this helps find the angle y such that sec(y) = -2.
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Graphs of Secant and Cosecant Functions
Relationship Between Secant and Cosine
Secant is the reciprocal of cosine, so sec(y) = 1/cos(y). To find y when sec(y) = -2, we rewrite it as cos(y) = -1/2. This relationship allows us to use cosine values to determine the angle.
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6:22Graphs of Secant and Cosecant Functions
Exact Values of Cosine for Special Angles
Certain angles have well-known cosine values, such as cos(120°) = cos(2π/3) = -1/2. Recognizing these exact values enables finding the precise angle y without a calculator when given sec(y) = -2.
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6:04Example 1
Related Practice
Textbook Question
Find the exact value of each real number y if it exists. Do not use a calculator.
y = sin⁻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
Find the exact value of each real number y. Do not use a calculator.
y = arccot (―1)
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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 θ = ―√2/2
Textbook Question
Solve each equation for x, where x is restricted to the given interval.
y = 6 cos x/4 , for x in [0, 4π]