Trigonometric Identities and Equations: Solutions and Methods

Chapter 11: Trigonometric Identities and Equations 11.1 Trigonometric Identities 11.2 Addition and Subtraction Formulas 11.3 Double-Angle, Half-Angle, and Product-Sum Formulas 11.4 Inverse Trigonometric Functions 11.5 Trigonometric Equations

11.5 Trigonometric Equations Solving a Trigonometric Equation by Linear Methods Solve 2 sin x 1 = 0 over the interval [0, 2 ). Example Analytic Solution Since this equation involves the first power of sin x, it is linear in sin x. 0 1 sin 2 = x However, if we do not restrict the domain there will be an infinite amount of answers since: x 6 = x x 2 0 Two x values that satisfy sin x = for are 6 6 5 = = and x x 2 sin 1 1 = sin x 2 6 5 = + k 2 = + k 2 and x for k any integer.

11.5 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 and6 . 5

11.5 Equations Solvable by Factoring Solve 2sin2x sin x 1 = 0 Example Solution This equation is of quadratic form so: sin sin 2 + x = 1 0 2 x x ) 1 = 2 ( sin )(sin 1 0 x = sin 1 x 0 x + = 2 sin 1 0 x or = sin 1 1 = sin x 2 7or 11 . The solutions for sin x = - in [0, 2 ) are x = 6 6 . The solutions for sin x = 1 in [0, 2 ) is x = 2 Thus the solutions are: 7 = 11 6 2 + = + k = + k 2 , 2 , 2 x k x x 6

11.5 Solving a Trigonometric Equation by Factoring Example Solve sin x tan x = sin x. Solution sin tan sin x x = sin tan sin x x x = 0 x x x ) 1 = sin (tan 0 = sin = or tan 1 x 0 0 x x = tan 1 5 = = = = 0 or or x x x x 4 + 4 = and = for solutions The are any integer x k x k k 4 Caution Avoid dividing both sides by sin x. The two solutions that make sin x = 0 would not appear.

11.5 Solving a Trigonometric Equation by Squaring and Trigonometric Substitution Example Solve over the interval [0, 2 ). Solution Square both sides and use the identity 1 + tan2x = sec2x. ) ( sec 3 tan x x x x x sec 3 tan 3 2 tan = + + + = tan 3 sec x x ( )2 2 + = Possible solutions are: and 6 5 11 2 2 . 6 + + = + tan 2 3 tan 3 1 tan 2 2 x x x = 1 Or are they? Check answers! 2 tan 3 tan = x 2 x 3

11.5 Trigonometric Equations Solving an Equation Using a Double-Number Identity Example Solve cos 2x = cos x over the interval [0, 2 ). Analytic Solution x cos 2 cos x 1 cos 2 1 cos cos 2 x x ) 1 )(cos 1 cos 2 ( + x x = = = = x x cos 0 0 2 2 + x = = = 2 cos cos 1 = 0 1 or or cos cos 1 1 0 x x x 2 2 4 , 0 { , }. Solving each equation yields the solution set 3 3

11.5 Solving an Equation Using a Double- Number Identity Graphical Solution Graph y = cos 2x cos x in an appropriate window, and find the x-intercepts. The x-intercept displayed is 2.0943951, an approximation for 2 /3. The other two correspond to 0 and 4 /3.

11.5 Solving an Equation Using a Double-Number Identity Example Solve 4 sin x cos x = over the interval [0, 2 ). 3 From the given domain for x, 0 x < 2 , the domain for 2x is 0 2x < 4 . Solution sin 4 sin 2 ( 2 sin 2 = x 3 cos x = x 3 = x x cos ) 3 2 3 Since 2 sin x cos x = sin 2x. = sin 2 x 2 3 = sin 2 x 2 2 7 8 = 2 , , , x 3 3 3 3 7 4 = , , , x 6 3 6 3

11.5 Solving an Equation that Involves Squaring Both Sides Example Solve tan 3x + sec 3x = 2 Solution Since the tangent and secant functions are related by the identity 1 + tan2 = sec2 , we begin by expressing everything in terms of secant. 2 3 sec 3 tan = + x x x x 3 sec 2 3 tan = x x 3 sec 4 4 3 tan = x sec 4 4 1 3 sec = + 3 2 2 sec x + 3 x 2 Square both sides. 2 Replace tan2 3x with sec2 3x 1. sec 3 x

11.5 Solving an Equation that Involves Squaring Both Sides + = x 3 sec 4 5 0 = 5 3 sec = x 2 2 sec 3 1 4 4 sec 3 sec 3 x x x All sol. are of the form 4 = + k 1 3 cos 2 x 4 5 5 4 1 = cos 3 4 4 x + k 1 cos 2 5 = x = cos 3 x 3 5 4 4 1 cos = 1 3 cos x 2 5 5 = + k x 3 3