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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 6.3.15
Capitolo 7, Problema 6.3.15

Answer each question.


Suppose solving a trigonometric equation for solutions over the interval [0°,360°) leads to 3θ = 180°, 630°, 720°,930°. What are the corresponding values of θ?

Guida verificata passo dopo passo
1
Identify the given equation: 3\(\theta\) equals the angles 180°, 630°, 720°, and 930°.
To find \(\theta\), divide each angle by 3, since the equation is 3\(\theta\) = angle.
Calculate \(\theta\) for each angle: \(\theta\) = \(\frac{180°}{3}\), \(\frac{630°}{3}\), \(\frac{720°}{3}\), \(\frac{930°}{3}\).
Simplify each fraction to get the values of \(\theta\).
Check that each \(\theta\) value lies within the interval [0°, 360°). If any value is outside this range, adjust it by subtracting or adding 360° as needed to bring it within the interval.

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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 often requires isolating the variable and considering the periodic nature of trigonometric functions to identify multiple solutions.
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How to Solve Linear Trigonometric Equations

Angle Division and Interval Constraints

When an equation involves a multiple of an angle, such as 3θ, dividing the solutions by that multiple gives the values of θ. It is important to adjust the solutions to fit within the original interval, here [0°, 360°), by considering the periodicity and reducing angles modulo 360° if necessary.
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Coterminal Angles

Understanding Angle Measures and Periodicity

Angles in trigonometry are often measured in degrees and repeat every 360°. Recognizing that angles differing by full rotations (multiples of 360°) represent the same position on the unit circle helps in identifying equivalent solutions and restricting answers to the given interval.
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Period of Sine and Cosine Functions
Pratica correlata
Domanda del libro di testo

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√3 sin x/2 = 3

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Domanda del libro di testo

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

sin θ/2 = 0

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Domanda del libro di testo

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.

3 csc x― 2√3 = 0

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Domanda del libro di testo

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.

5 + 5 tan² θ = 6 sec θ

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Domanda del libro di testo

Answer each question.


Suppose solving a trigonometric equation for solutions over the interval [0, 2π) leads to 2x = 2π/3, 2π, 8π/3. What are the corresponding values of x?

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Domanda del libro di testo

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/2) = √2 ― sin (x/2)

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