Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Problema 6.3.9

Capitolo 7, Problema 6.3.9

Solve for exact solutions over the interval [0°, 360°).
cos θ/2 = -1/2

Guida verificata passo dopo passo

Start by rewriting the equation \( \cos \frac{\theta}{2} = -\frac{1}{2} \) and recognize that you need to find all angles \( \frac{\theta}{2} \) whose cosine value is \( -\frac{1}{2} \).

Recall the unit circle values where \( \cos x = -\frac{1}{2} \). These occur at angles \( x = 120^\circ \) and \( x = 240^\circ \) within one full rotation \( [0^\circ, 360^\circ) \).

Set \( \frac{\theta}{2} = 120^\circ + 360^\circ k \) and \( \frac{\theta}{2} = 240^\circ + 360^\circ k \), where \( k \) is any integer, to account for all possible solutions.

Multiply both sides of each equation by 2 to solve for \( \theta \): \( \theta = 240^\circ + 720^\circ k \) and \( \theta = 480^\circ + 720^\circ k \).

Finally, find all values of \( \theta \) within the interval \( [0^\circ, 360^\circ) \) by substituting integer values of \( k \) and selecting those that fall within the given range.

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Solving Trigonometric Equations

Solving trigonometric equations involves finding all angle values that satisfy the given equation within a specified interval. This requires isolating the trigonometric function and determining the angles whose function values match the given number, considering the periodic nature of trigonometric functions.

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How to Solve Linear Trigonometric Equations

Cosine Function and Its Values

The cosine function relates an angle to the x-coordinate on the unit circle. Knowing key cosine values, such as cos 60° = 1/2 and cos 120° = -1/2, helps identify angles that satisfy equations like cos(θ/2) = -1/2. Understanding cosine’s behavior over one full rotation (0° to 360°) is essential.

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Sine, Cosine, & Tangent of 30°, 45°, & 60°

Interval and Angle Restrictions

When solving trigonometric equations over a specific interval, it is important to consider the domain of the original variable. Here, θ is restricted to [0°, 360°), so after solving for θ/2, solutions must be adjusted and filtered to ensure θ falls within this range.

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Coterminal Angles

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