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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 6.3.37
Chapter 7, Problem 6.3.37

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/2 = 1

Verified step by step guidance
1
Identify the given equation: \(\cos \frac{\theta}{2} = 1\). We want to find all values of \(\theta\) (in degrees) that satisfy this equation.
Recall that \(\cos x = 1\) when \(x = 2k\pi\) for any integer \(k\), where \(x\) is in radians. Since the argument here is \(\frac{\theta}{2}\), set \(\frac{\theta}{2} = 2k\pi\).
Solve for \(\theta\) by multiplying both sides by 2: \(\theta = 4k\pi\). This gives the general solution in radians.
Convert the general solution to degrees by using the conversion \(180^\circ = \pi\) radians: \(\theta = 4k\pi \times \frac{180^\circ}{\pi} = 720k^\circ\).
Write the least possible nonnegative angle measures by choosing integer values of \(k\) starting from 0, which gives \(\theta = 0^\circ, 720^\circ, 1440^\circ, \ldots\). Since angles are often expressed within \(0^\circ\) to \(360^\circ\), the principal solution is \(\theta = 0^\circ\).

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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

Solving trigonometric equations involves finding all angle values that satisfy the given equation. For cosine equations like cos(θ/2) = 1, you identify angles where the cosine function equals 1, then solve for θ considering the function's periodicity.
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How to Solve Linear Trigonometric Equations

Understanding Angle Measures and Units

Angles can be measured in degrees or radians, and converting between these units is essential. The problem requires solutions in both radians and degrees, so knowing that 180° equals π radians helps in converting and expressing answers correctly.
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Reference Angles on the Unit Circle

Using the Unit Circle and Periodicity of Cosine

The unit circle shows cosine values for angles around a circle, with cos(θ) = 1 at θ = 0° (or 0 radians) and repeating every 360° (2π radians). Recognizing this periodicity allows finding all solutions by adding multiples of the period to the principal solution.
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Sine, Cosine, & Tangent on the Unit Circle
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.

6 sin² θ + sin θ = 1

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.

sin x (3 sin x - 1) = 1

Textbook Question

Solve for exact solutions over the interval [0°, 360°).

sin θ/2 = -√3/2

Textbook Question

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.

9 sin² θ ― 6 sin² θ = 1

Textbook Question

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.

sin x = sin 2x

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 sin θ = 2 cos 2θ