Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Problem 11

Chapter 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 guidance

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.

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.

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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Example 1

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°).

<IMAGE>

cos θ = ―1/2

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Textbook Question

Find the exact value of each real number y if it exists. Do not use a calculator.

y = sin⁻¹ 0

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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°).

<IMAGE>

sin θ = ―√2/2

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Textbook Question

Solve each equation for x, where x is restricted to the given interval.

y = ― 2 cos 5x , for x in [0, π/5]

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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]

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