742 CHAPTER 7 Trigonometric Identities and Equations To find all solutions, we add integer multiples of the period of the tangent function, which is p, to each solution found previously. Although not unique, a common form of the solution set of the equation, written using the least possible nonnegative angle measures, is given as follows. 52.8476 + np, 1.2768 + np, where n is any integer6 Round to four decimal places. S Now Try Exercise 57. EXAMPLE 4 Solving aTrigonometric Equation (Quadratic Formula) Solve cot x1cot x + 32 = 1 for all solutions. SOLUTION We multiply the factors on the left and subtract 1 to write the equation in standard quadratic form. cot x1cot x + 32 = 1 Original equation cot2 x + 3 cot x - 1 = 0 Distributive property; Subtract 1. This equation is quadratic in form but cannot be solved using the zero-factor property. Therefore, we use the quadratic formula, with a = 1, b = 3, c = -1, and cot x as the variable. cot x = -b {2b2 - 4ac 2a Quadratic formula = -3 {232 - 41121-12 2112 a = 1, b = 3, c = -1 = -3 {29 + 4 2 Simplify. = -3 { 13 2 Add under the radical. cot x ≈ -3.302775638 or cot x ≈0.3027756377 Use a calculator. x ≈cot-11-3.3027756382 or x ≈cot-110.30277563772 Definition of inverse cotangent x ≈tan-1 a 1 -3.302775638b + p or x ≈tan-1 a 1 0.3027756377b Write inverse cotangent in terms of inverse tangent. x ≈ -0.2940013018 + p or x ≈1.276795025 Use a calculator in radian mode. x ≈2.847591352 Be careful with signs. Trigonometric Identity Substitutions Recall that squaring each side of an equation, such as 2 x + 4 = x + 2, will yield all solutions but may also give extraneous solutions — solutions that satisfy the final equation but not the original equation. As a result, all proposed solutions must be checked in the original equation as shown in Example 5. LOOKING AHEAD TO CALCULUS There are many instances in calculus where it is necessary to solve trigonometric equations. Examples include solving related-rates problems and optimization problems.
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