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 6.2.47

Chapter 7, Problem 6.2.47

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.
cos θ + 1 = 0

Verified step by step guidance

Start with the given equation: \(\cos \theta + 1 = 0\).

Isolate the cosine term by subtracting 1 from both sides: \(\cos \theta = -1\).

Recall that \(\cos \theta = -1\) at specific angles on the unit circle. Identify all angles \(\theta\) where the cosine value is exactly \(-1\) within one full rotation (0° to 360° or 0 to \(2\pi\) radians).

Express the general solution for \(\theta\) in degrees and radians. Since cosine is periodic with period \(360^\circ\) or \(2\pi\) radians, write the solutions as \(\theta = 180^\circ + 360^\circ k\) or \(\theta = \pi + 2\pi k\), where \(k\) is any integer.

For the least possible nonnegative angle measure, provide the principal solution(s) within the interval \([0, 360^\circ)\) or \([0, 2\pi)\), and mention the general solution to cover all possible angles.

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Key Concepts

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

Solving Basic Trigonometric Equations

This involves isolating the trigonometric function and finding all angle values that satisfy the equation within a given domain. For example, solving cos θ + 1 = 0 means finding θ where cos θ = -1, which occurs at specific standard angles.

Angle Measurement and Conversion

Understanding the difference between radians and degrees is essential, as the problem requires solutions in both units. Knowing how to convert between radians and degrees (1 radian = 180/π degrees) helps in expressing answers correctly and rounding them as specified.

General Solutions and Principal Values

Trigonometric equations often have infinitely many solutions due to periodicity. The general solution expresses all possible angles, while principal values are the smallest nonnegative angles within one full rotation (0 to 2π radians or 0° to 360°). Writing answers using least possible nonnegative measures ensures clarity.

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Related Practice

Textbook Question

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

2 cos² x + cos x ― 1 = 0

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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.

2 sin 2x ― x³ + 1 = 0

√2 sin 3x - 1 = 0

Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate

tan 2x + sec 2x = 3

3 csc² x/2 = 2 sec x

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

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Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.cos θ + 1 = 0

Verified video answer for a similar problem:

Key Concepts

Solving Basic Trigonometric Equations

Angle Measurement and Conversion

General Solutions and Principal Values

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.
cos θ + 1 = 0