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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 9
Chapter 7, Problem 9

Find the exact value of each real number y. Do not use a calculator.
y = cos⁻¹ (―√2/2)

Verified step by step guidance
1
Recognize that the problem asks for the angle \( y \) such that \( \cos(y) = -\frac{\sqrt{2}}{2} \). This means we need to find the angle whose cosine value is \( -\frac{\sqrt{2}}{2} \).
Recall the unit circle values for cosine. The value \( \frac{\sqrt{2}}{2} \) corresponds to angles \( \frac{\pi}{4} \) and \( \frac{7\pi}{4} \) in the first and fourth quadrants, respectively. Since the cosine is negative, \( y \) must be in the second or third quadrant where cosine values are negative.
Identify the angles in the second and third quadrants where cosine equals \( -\frac{\sqrt{2}}{2} \). These are \( \frac{3\pi}{4} \) (second quadrant) and \( \frac{5\pi}{4} \) (third quadrant).
Recall that the range of the inverse cosine function \( \cos^{-1}(x) \) is \( [0, \pi] \), which means the output angle \( y \) must lie between 0 and \( \pi \).
Therefore, select the angle within the range \( [0, \pi] \) where \( \cos(y) = -\frac{\sqrt{2}}{2} \), which is \( y = \frac{3\pi}{4} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Cosine Function (cos⁻¹ or arccos)

The inverse cosine function returns the angle whose cosine is a given number. It is defined for inputs between -1 and 1 and outputs angles in the range [0, π] radians. Understanding this helps find the angle y such that cos(y) equals the given value.
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Inverse Cosine

Exact Values of Cosine for Special Angles

Certain angles have well-known cosine values expressed in terms of square roots, such as cos(π/4) = √2/2. Recognizing these exact values allows you to identify the angle corresponding to a given cosine value without a calculator.
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Example 1

Handling Negative Cosine Values

Cosine is negative in the second quadrant (angles between π/2 and π). Since arccos outputs angles in [0, π], a negative cosine value corresponds to an angle in this range, typically π minus the reference angle with positive cosine.
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Sine, Cosine, & Tangent of 30°, 45°, & 60°
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°).                     

<IMAGE>

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 = sec⁻¹ (―2)

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

Textbook Question

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

y = tan⁻¹ (―√3)