SECTION 7.3 Trigonometric Equations 493 In this section, we discuss trigonometric equations —that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or, possibly, are not satisfied by any values of the variable). The values that satisfy the equation are called solutions of the equation. 1 Solve Equations Involving a Single Trigonometric Function 7.3 Trigonometric Equations Now Work the ‘Are You Prepared?’ problems on page 498. PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Solve Equations Involving a Single Trigonometric Function (p. 493) 2 Solve Trigonometric Equations Using a Calculator (p. 496) 3 Solve Trigonometric Equations Quadratic in Form (p. 496) 4 Solve Trigonometric Equations Using Fundamental Identities (p. 497) 5 Solve Trigonometric Equations Using a Graphing Utility (p. 498) Checking Whether a Given Number Is a Solution of a Trigonometric Equation Determine whether θ π = 4 is a solution of the equation θ − = 2sin 1 0. Is θ π = 6 a solution? Solution EXAMPLE 1 Replace θ by π 4 in the equation θ − = 2 sin 1 0. The result is π − = ⋅ − = − ≠ 2 sin 4 1 2 2 2 1 2 1 0 Therefore, π 4 is not a solution. Next replace θ by π 6 in the equation. The result is π − = ⋅ − = 2 sin 6 1 2 1 2 1 0 Therefore, π 6 is a solution of the equation θ − = 2 sin 1 0. The equation θ − = 2sin 1 0 in Example 1 has other solutions besides θ π = 6 . For example, θ π = 5 6 is also a solution, as is θ π = 13 6 . (Check this for yourself.) In fact, the equation has an infinite number of solutions because of the periodicity of the sine function, as can be seen in Figure 25, which shows the graph of = − y x 2 sin 1 using a TI-84 Plus. Each x -intercept of the graph represents a solution to the equation − = x 2 sin 1 0. Unless the domain of the variable is restricted, we need to find all the solutions of a trigonometric equation.As the next example illustrates, finding all the solutions can be accomplished by first finding solutions over an interval whose length equals the period of the function and then adding multiples of that period to the solutions found. Figure 25 = − y x 2 sin 1 22p 24 4p 2 Y1 5 2sin x 2 1 • Linear Equations (Section A.6, pp. A45–A47) • Values of the Trigonometric Functions (Section 6.2, pp. 398–405) • Quadratic Equations (Section A.6, pp. A47–A53) • Equations Quadratic in Form (Section A.6, pp. A53–A54) • Using a Graphing Utility to Solve Equations (Section 1.4, pp. 28–31) • Fundamental Identities (Section 6.3, pp. 417–419)
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