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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 1 - FunctionsProblema 54
Capitolo 1, Problema 54

For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.


cos 2θ + cos θ = 0

Guida verificata passo dopo passo
1
Start by using the double angle identity for cosine: \( \cos 2\theta = 2\cos^2\theta - 1 \). Substitute this into the equation to get \( 2\cos^2\theta - 1 + \cos\theta = 0 \).
Rearrange the equation to form a quadratic in terms of \( \cos\theta \): \( 2\cos^2\theta + \cos\theta - 1 = 0 \).
Let \( x = \cos\theta \). The equation becomes \( 2x^2 + x - 1 = 0 \). Solve this quadratic equation using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 1 \), and \( c = -1 \).
Calculate the discriminant \( b^2 - 4ac \) and find the roots \( x_1 \) and \( x_2 \). These roots represent the possible values for \( \cos\theta \).
For each root, determine the corresponding angle \( \theta \) within the interval \( 0 \leq \theta \leq 2\pi \) by considering the unit circle and the possible quadrants where cosine is positive or negative.

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable. In this problem, the double angle identity for cosine, cos(2θ) = 2cos²(θ) - 1, can be useful to simplify the equation. Understanding these identities is crucial for manipulating and solving trigonometric equations.
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Verifying Trig Equations as Identities

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy the equation within a specified interval. In this case, we need to find values of θ that make the equation cos(2θ) + cos(θ) = 0 true, specifically within the range 0 ≤ θ ≤ 2π. This often requires using algebraic techniques and understanding the properties of trigonometric functions.
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Solving Logarithmic Equations

Unit Circle

The unit circle is a fundamental concept in trigonometry that helps visualize the values of sine and cosine for different angles. It provides a geometric interpretation of trigonometric functions, where the x-coordinate represents cos(θ) and the y-coordinate represents sin(θ). Understanding the unit circle is essential for determining the angles that correspond to specific trigonometric values, especially when solving equations.
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Evaluate Composite Functions - Values on Unit Circle
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