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5.5 Solving Trigonometric Equations Example 1 A) Is a solution to ? B) Is a solution to cos x = sin 2x ?

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Solving Trigonometric Equations - Overview

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Trigonometric Equations with a Single Trig Function For equations with a single trig function, isolate the trig function on one side. Solve for the variable by identifying the appropriate angles. Be prepared to express your answer in radian measure.

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Example 2 Find all solutions for

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Example 2 - Solution where n is any integer

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Example 3 Solve the equation on the interval [0º, 360º) sin x = x = 30º, 150º

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Other Strategies for Solving Solving Trig Equations Put the equation in terms of one trig function (if possible). Solve for the trig function (using algebra – addition, subtraction, multiplication, division, factoring). Solve for the variable (using inverse trig functions, reference angles). Use a fundamental identity to end up with a single trig function.

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Example 4 To solve an equation containing a single trig function: Solve: 3sinx – 2 = 5sinx - 1 * Isolate the function on one side of the equation. * Solve for the variable. Solution: 3sinx - 5sinx = -1 +2 -2sinx = 1 sinx = -1/2 (Remember: x are the angles whose sine is -1/2)

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Example 5 Solve the equation on the interval [0, 2π) 2 cos x − 1 = 0 2 cos x = 1 cos x = x =

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Example 6 -Trigonometric Equations Quadratic in Form. Ans. π/6, π/2, 5π/6 Try to solve by factoring It factors in the same manner as = (2x -1)(x – 1) Solution: (2sinx – 1)(sinx -1) = 0 2sinx – 1 = 0 2sinx = 1 sinx = ½ Therefore x = π/6, 5π/6 sinx – 1 = 0 sinx = 1 x = π/2

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Example 8: Solve an Equation with a Multiple Angle.

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Ans. x = Example 9 - Multiple Angle

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Ans. 0, π Move all terms to one side, then factor out a common trig function. Example 10

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Ans. π/3, 5π/3 The equation contains more than one trig function; there is no common trig function. Try using an identity. Example 11

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Example 12 Solve the equation: cos2x + 3sinx – 2 = 0, 0 ≤ x ≤ 2π Ans. π/6, π/2, and 5π/6

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Example 13 Solve the equation: sinx cosx= -1/2, 0 ≤ x ≤ 2π Ans. 3π/4, 7π/4

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Example 14 - using a calculator to solve Solve the equation correct to four decimal places, 0 ≤ x ≤ 2π a. tan x = 3.1044 b. sin x = -0.2315 Ans. a. 1.2592, 4.4008 b. 3.3752, 6.0496 Use a calculator to find the reference angle, then use your knowledge of signs of trigonometric functions to find x in the required interval.

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Ans. 2.3423, 3.9409 The equation is in quadratic form, but does not factor. Use the quadratic formula to solve for the trig function of x, then use a calculator and the Example 15

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