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Copyright © 2011 Pearson Education, Inc. Slide 9.5-1 Chapter 9: Trigonometric Identities and Equations (I) 9.1 Trigonometric Identities 9.2 Sum and Difference Identities 9.3 Further Identities 9.4 The Inverse Circular Functions 9.5 Trigonometric Equations and Inequalities (I) 9.6 Trigonometric Equations and Inequalities (II) Copyright © 2011 Pearson Education, Inc. Slide 9.5-2 9.5 Trigonometric Equations and Inequalities (I) • Solving a Trigonometric Equation by Linear Methods Example Solve 2 sin x – 1 = 0 over the interval [0, 2). Analytic Solution Since this equation involves the first power of sin x, it is linear in sin x. 2 sin x 1 0 2 sin x 1 1 sin x 2 Two values for x, 6 and 56 , satisfy sin x 12 for 0 x 2 . Therefore the solution set is 6 , 56 . Copyright © 2011 Pearson Education, Inc. Slide 9.5-3 9.5 Solving a Trigonometric Equation by Linear Methods Graphing Calculator Solution Graph y = 2 sin x – 1 over the interval [0, 2]. The x-intercepts have the same decimal approximations as 6 and 56 . Copyright © 2011 Pearson Education, Inc. Slide 9.5-4 9.5 Solving Trigonometric Inequalities Example Solve for x over the interval [0, 2). (a) 2 sin x –1 > 0 and (b) 2 sin x –1 < 0. Solution (a) Identify the values for which the graph of y = 2 sin x –1 is above the x-axis. From the previous graph, the solution set is 6 , 56 . (b) Identify the values for which the graph of y = 2 sin x –1 is below the x-axis. From the previous graph, the solution set is 0, 6 56 ,2 . Copyright © 2011 Pearson Education, Inc. Slide 9.5-5 9.5 Solving a Trigonometric Equation by Factoring Example Solve sin x tan x = sin x over the interval [0°, 360°). Solution sin x tan x sin x sin x tan x sin x 0 sin x(tan x 1) 0 sin x 0 or tan x 1 0 tan x 1 x 0 or x 180 x 45 or x 225 The solution set is 0 ,45 ,180 ,225 . Caution Avoid dividing both sides by sin x. The two solutions that make sin x = 0 would not appear. Copyright © 2011 Pearson Education, Inc. Slide 9.5-6 9.5 Equations Solvable by Factoring Example Solve tan2 x + tan x –2 = 0 over the interval [0, 2). Solution This equation is quadratic in term tan x. tan x tan x 2 0 (tan x 1)(tan x 2) 0 tan x 1 0 tan x 2 0 or tan x 1 tan x 2 or 2 The solutions for tan x = 1 in [0, 2) are x = 4 or 54 . Use a calculator to find the solution to tan-1(–2) –1.107148718. To get the values in the interval [0, 2), we add and 2 to tan-1(–2) to get x = tan-1(–2) + 2.03443936 and x = tan-1(–2) + 2 5.176036589. Copyright © 2011 Pearson Education, Inc. Slide 9.5-7 9.5 Solving a Trigonometric Equation Using the Quadratic Formula Example Solve cot x(cot x + 3) = 1 over the interval [0, 2). Solution Rewrite the expression in standard quadratic form to get cot2 x + 3 cot x – 1 = 0, with a = 1, b = 3, c = –1, and cot x as the variable. 3 9 4 3 13 cot x 2 2 cot x 3.302775638 or cot x .3027756377 Since we cannot take the inverse cotangent with the calculator, we use the fact that cot x tan1 x . Copyright © 2011 Pearson Education, Inc. Slide 9.5-8 9.5 Solving a Trigonometric Equation Using the Quadratic Formula cot x 3.302775638 or tan x .3027756377 or cot x .3027756377 tan x 3.302775638 x 1.276795025 x .2940013018 or The first of these, –.29400113018, is not in the desired interval. Since the period of cotangent is , we add and then 2 to –.29400113018 to get 2.847591352 and 5.989184005. The second value, 1.276795025, is in the interval, so we add to it to get another solution. The solution set is {1.28, 2.85, 4.42, 5.99}. Copyright © 2011 Pearson Education, Inc. Slide 9.5-9 9.5 Solving a Trigonometric Equation by Squaring and Trigonometric Substitution Example Solve tan x 3 sec x over the interval [0, 2). Solution Square both sides and use the identity 1 + tan2 x = sec2 x. tan x 3 sec x 2 2 tan x 2 3 tan x 3 sec x 2 2 tan x 2 3 tan x 3 1 tan x 2 3 tan x 2 1 3 tan x 3 3 Possible solutions are 56 and 116 . Verify that the only solution is 116 . Copyright © 2011 Pearson Education, Inc. Slide 9.5-10 9.4 The Inverse Sine Function Solving a Trigonometric Equation Analytically 1. Decide whether the equation is linear or quadratic, so you can determine the solution method. 2. If only one trigonometric function is present, solve the equation for that function. 3. If more than one trigonometric function is present, rearrange the equation so that one side equals 0. Then try to factor and set each factor equal to 0 to solve. 4. If the equation is quadratic in form, but not factorable, use the quadratic formula. Check that solutions are in the desired interval. 5. Try using identities to change the form of the equation. It may be helpful to square each side of the equation first. If this is done, check for extraneous values. Copyright © 2011 Pearson Education, Inc. Slide 9.5-11 9.4 The Inverse Sine Function Solving a Trigonometric Equation Graphically 1. For an equation of the form f(x) = g(x), use the intersection-of-graphs method. 2. For an equation of the form f(x) = 0, use the x-intercept method. Copyright © 2011 Pearson Education, Inc. Slide 9.5-12