Domanda del libro di testo
In Exercises 69–74, rewrite each expression as a simplified expression containing one term. sin (α - β) cos β + cos (α - β) sin β
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 3, Problema 71
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). cos 2x = cos x
Guida verificata passo dopo passo1
Recall the double-angle identity for cosine: \(\cos 2x = 2\cos^{2} x - 1\). Substitute this into the equation to rewrite it as \(2\cos^{2} x - 1 = \cos x\).
Rearrange the equation to set it equal to zero: \(2\cos^{2} x - \cos x - 1 = 0\).
Let \(y = \cos x\) to transform the equation into a quadratic form: \(2y^{2} - y - 1 = 0\).
Solve the quadratic equation \(2y^{2} - y - 1 = 0\) for \(y\) using the quadratic formula \(y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=2\), \(b=-1\), and \(c=-1\).
For each solution \(y\), find the corresponding values of \(x\) in the interval \([0, 2\pi)\) by solving \(\cos x = y\). Use the unit circle or inverse cosine function to determine all valid solutions.
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Double-Angle Identity for Cosine
The double-angle identity expresses cos(2x) in terms of cos(x) and sin(x). Common forms include cos(2x) = 2cos²(x) - 1 or cos(2x) = 1 - 2sin²(x). This identity allows rewriting the equation cos(2x) = cos(x) into a solvable form involving a single trigonometric function.
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05:06
Double Angle Identities
Solving Trigonometric Equations
Solving trigonometric equations involves manipulating the equation using identities and algebraic techniques to isolate the variable. After rewriting, solutions are found by considering the unit circle values where the trigonometric functions equal specific values within the given interval.
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4:34How to Solve Linear Trigonometric Equations
Interval Restriction and General Solutions
When solving trigonometric equations, solutions must be restricted to the specified interval, here [0, 2π). This means identifying all angles within one full rotation of the unit circle that satisfy the equation, ensuring the final answer fits the problem’s domain.
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In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). cos 2x + 5 cos x + 3 = 0
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In Exercises 69–74, rewrite each expression as a simplified expression containing one term. cos (α + β) cos β + sin (α + β) sin β
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