Copyright © 2011 Pearson Education, Inc. Slide 9.5-1

Copyright © 2011 Pearson Education, Inc. Slide Chapter 9: Trigonometric Identities and Equations (I) 9.1Trigonometric Identities 9.2Sum and Difference Identities 9.3Further Identities 9.4The Inverse Circular Functions 9.5Trigonometric Equations and Inequalities (I) 9.6Trigonometric Equations and Inequalities (II)

Copyright © 2011 Pearson Education, Inc. Slide Trigonometric Equations and Inequalities (I) Solving a Trigonometric Equation by Linear Methods ExampleSolve 2 sin x – 1 = 0 over the interval [0, 2  ). Analytic SolutionSince this equation involves the first power of sin x, it is linear in sin x.

Copyright © 2011 Pearson Education, Inc. Slide 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

Copyright © 2011 Pearson Education, Inc. Slide Solving Trigonometric Inequalities ExampleSolve 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 (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

Copyright © 2011 Pearson Education, Inc. Slide Solving a Trigonometric Equation by Factoring ExampleSolve sin x tan x = sin x over the interval [0°, 360°). Solution 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 Equations Solvable by Factoring ExampleSolve tan 2 x + tan x –2 = 0 over the interval [0, 2  ). SolutionThis equation is quadratic in term tan x. The solutions for tan x = 1 in [0, 2  ) are x = Use a calculator to find the solution to tan -1 (–2)  – To get the values in the interval [0, 2  ), we add  and 2  to tan -1 (–2) to get x = tan -1 (–2) +   and x = tan -1 (–2) + 2  

Copyright © 2011 Pearson Education, Inc. Slide Solving a Trigonometric Equation Using the Quadratic Formula ExampleSolve cot x(cot x + 3) = 1 over the interval [0, 2  ). SolutionRewrite the expression in standard quadratic form to get cot 2 x + 3 cot x – 1 = 0, with a = 1, b = 3, c = –1, and cot x as the variable. Since we cannot take the inverse cotangent with the calculator, we use the fact that

Copyright © 2011 Pearson Education, Inc. Slide Solving a Trigonometric Equation Using the Quadratic Formula The first of these, – , is not in the desired interval. Since the period of cotangent is , we add  and then 2  to – to get and The second value, , 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 Solving a Trigonometric Equation by Squaring and Trigonometric Substitution ExampleSolve over the interval [0, 2  ). SolutionSquare both sides and use the identity 1 + tan 2 x = sec 2 x.

Copyright © 2011 Pearson Education, Inc. Slide 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 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.