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5.3 Trigonometric Equations: An Algebraic Approach

5.3 Trigonometric Equations: An Algebraic Approach. Introduction Solving trigonometric equations using an algebraic approach. Solving a Simple Sine Equation. Find all solutions in the unit circle to sin x = 1/ √2. Solution: Use the unit circle to determine that one solution is x = p /4.

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5.3 Trigonometric Equations: An Algebraic Approach

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  1. 5.3 Trigonometric Equations:An Algebraic Approach • Introduction • Solving trigonometric equations using an algebraic approach

  2. Solving a Simple Sine Equation • Find all solutions in the unit circle to sin x = 1/√2. • Solution: • Use the unit circle to determine that one solution is x = p/4. • It can be seen that another point on the circle with the desired height is x = 3p/4.

  3. Suggestions for Solving Trigonometric Equations

  4. Exact Solutions Using Factoring • Example: Find all solutions in [0, 2p] to 2 sin2x + sin x = 0 • Solution: • 2 sin2x + sin x = 0 • sin x(2 sin x + 1) = 0 • sin x = 0 or sin x = -1/2 • Find these values on the unit circle. • The solutions are x = 0, p, 7p/6, and 11p/6.

  5. Exact Solutions Using Identities and Factoring • Example: Find all solutions for sin 2x = sin x, 0  x  2p. • Solution: • sin 2x = sin x • 2 sin x cos x = sin x • 2 sin x cos x – sin x = 0 • sin x (2 cos x – 1) = 0 • sin x = 0 or cos x = ½ • From the unit circle we find 4 solutions: x = 0, p/3, p, and 5p/3.

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