Solving Quadratic Trigonometric Equations

Transcription

1 Solving Quadratic Trigonometric Equations 6.6 SKILL BUILDER Often in mathematics, it is necessar to combine skills and strategies ou have used before to deal with more complicated problems. A quadratic trigonometric equation is a trigonometric equation with sin x, cos x, or tan x, and has as its highest power. For instance, sin x sin x is a quadratic trigonometric equation. These equations are new to ou, but ou can solve them b combining methods for solving quadratic equations and linear trigonometric equations. Solving b Factoring Recall that factoring can help solve some quadratic equations. Example Solve x x 5. x x 5 Express the equation in the form ax bx c 0. x x 5 0 Factor. (x 5)(x 3) 0 Set each factor equal to 0 and solve. x 5 0 and x 3 0 x 5 and x 3 This strateg can be applied to quadratic trigonometric equations. Factoring to Solve Quadratic Trigonometric Equations Factoring can be used to solve some quadratic trigonometric equations once the equation has been expressed in standard form. Example Solve sin x, 0 x p. sin x Rearrange the equation so one side equals 0. sin x 0 Factor. (sin x )(sin x ) 0 Set each factor equal to 0 and solve. sin x 0 and sin x 0 sin x and sin x 6.6 SOLVING QUADRATIC TRIGONOMETRIC EQUATIONS 54

2 In the graph of sin x, notice that when x p, and when x 3 p. 0 = sin v 3 v sin p and sin 3 p - Therefore, x p and 3 p. Using Special Triangles Sometimes the special triangles can be used. Example 3 Solve cos v cos v, 0 v p. cos v cos v Rearrange so one side equals 0. cos v cos v 0 Factor. ( cos v )(cos v ) 0 Set each factor equal to 0 and solve. cos v 0 and cos v 0 cos v and cos v cos v The equation cos v can be solved using the p 3 p 6 special triangle. Cosine is positive in quadrants I and IV. In this case, p 3 is the related angle. In quadrant I: x, r v p 3 p v (3) () v p 6 r = x = 3 = 3 x In quadrant IV: x, r v 5 3p v (3 5 v 5 6p p )() 3 r = x = v = 3 = 5 3 x = 3 However, the period of the equation is p. Add p to the solutions to find all the solutions in the domain. v p 6 v p 6 p v 3 5 p v p p 6 7 p 6 6p 54 CHAPTER 6 EXTENDING SKILLS WITH TRIGONOMETRY

3 Use the graph of cos v to solve v. 0 = cos v 3 v v p v p - However, the period of the equation is p. Add p to the solutions to find all the solutions in the domain. v 5 p v 6 p p 3 p Therefore, cos v cos v, 0 v p, has six solutions. v p 6 v p v 3 5 p v p v p v 6 6p Using a Calculator Often ou must use the Pthagorean identit and a calculator to solve an equation. Example 4 Solve 8 3 sin x cos x, 0 x sin x cos x Rearrange so one side equals 0. cos x 3 sin x 8 0 Express the equation in terms of sin x (cos x sin x). ( sin x) 3 sin x 8 0 Expand. sin x 3 sin x 8 0 Simplif. sin x 3 sin x 4 0 Factor. (3 sin x 4)(4 sin x ) 0 Set each factor equal to 0 and solve. 3 sin x 4 0 and 4 sin x 0 3 sin x 4 and 4 sin x sin x 4 3 and sin x 4 The equation sin x 4 3 has no solutions, since sin x. The equation sin x 4 has two solutions, and x is an angle in quadrants I or II. Use a scientific or graphing calculator. 6.6 SOLVING QUADRATIC TRIGONOMETRIC EQUATIONS 543

4 x sin 4 x 4.5 The angle in quadrant II has 4.5 as its related angle. Therefore, x or The equation 8 3 sin x cos x, 0 x 360 has two solutions, x 4.5 and x x = = x Using Graphing Technolog Graphing technolog can be used to solve a quadratic trigonometric equation. This approach will work even when the equation cannot be factored. Example 5 Solve 6 sin x sin x 0, 0 x p, using graphing technolog. Find the solutions to this equation b graphing 6 sin x sin x over the domain 0 x p and locating the zeros, or x-intercepts.. Put the calculator in radian mode. On the TI-83 Plus, press z. Then scroll down to Radian and press Õ.. Enter the relation into the equation editor. 3. Set the window for the given domain. Xmin 0, Xmax p, and Xscl p/. 4. Graph b using zoom fit. Press q A. 5. Determine all the zeros. Press r, then choose the left bound and right bound. Press Õ on Guess. 544 CHAPTER 6 EXTENDING SKILLS WITH TRIGONOMETRY

5 6. The three other zeros are found similarl. The equation 6 sin x sin x 0 has solutions x 0.536, x.680, x 3.873, and x , 0 x p. Ke Ideas An trigonometric equation with sin x, cos x, or tan x, with as its highest power, is called a quadratic trigonometric equation. For example, sin x sin x is a quadratic trigonometric equation. A quadratic trigonometric equation must be expressed in terms of the same trigonometric ratio before it can be solved. Factoring often helps when solving a quadratic expression. Once the expression has been factored, set each factor equal to 0 to obtain two linear trigonometric equations that can be solved. It is possible that one of the factors ma not lead to a solution of the original equation. This occurs when sin v >, cos v >, sin v <, or cos v <. The period and the domain of the corresponding functions must be considered when determining all possible solutions. A close approximation to the solution for an quadratic trigonometric equation can be found b graphing the corresponding trigonometric function using graphing technolog and determining the zeros of the function over the given domain. Practise, Appl, Solve 6.6 A. Factor. (a) 5x 0x (b) x 3x 40 (c) 0x x 6 (d) x 8. Factor. (a) sin v sin v (b) cos v cos v (c) 3 sin v sin v (d) 4 cos v 3. Verif that the value of v is a solution of each equation. (a) sin v, v 3 4p (b) 4 cos p v 3, v (c) sin v sin v, v p 6.6 SOLVING QUADRATIC TRIGONOMETRIC EQUATIONS 545

6 4. Solve each equation for x, 0 x 360. (a) sin x cos x 0 (b) sin x (cos x ) 0 (c) (sin x ) cos x 0 (d) cos x ( sin x 3 ) 0 (e) ( sin x )( sin x ) 0 (f) (sin x )(cos x ) 0 5. Solve each equation for x, 0 x p. (a) ( sin x ) cos x 0 (b) (sin x ) 0 (c) ( cos x 3 ) sin x 0 (d) ( cos x )( sin x 3 ) 0 (e) ( cos x )( cos x ) 0 (f) (sin x )(cos x ) 0 B 6. Solve for v to the nearest degree, 0 v 360. (a) sin v (b) cos v (c) tan v (d) 4 cos v (e) 3 tan v (f) sin v 7. (a) Write sin x sin x in factored form. (b) Use the factors from (a) to solve sin x sin x 0, 0 x p. 8. (a) Write cos x cos x in factored form. (b) Use the factors in (a) to solve cos x cos x 0, 0 x Solve for x to the nearest degree, 0 x 360. (a) sin x sin x 0 (b) cos x cos x (c) tan x tan x 3 0 (d) 6 sin x sin x (e) cos x 6 cos x 5 0 (f) 4 sin x 3 sin x 0. Solve for v to the nearest hundredth of a radian, 0 v p. (a) cos v cos v 0 (b) sin v sin v (c) cos v cos v (d) sin v 5 sin v 3 0 (e) 3 tan v tan v (f) sin v sin v 6 0. Solve for x to the nearest degree, 0 x 360. (a) cos x cos x (b) sin x. Solve for v to the nearest hundredth of a radian, 0 v p. (a) cos v sin v (b) sin v cos v 0 (c) sin cos v (d) 3 5 sin v cos v CHAPTER 6 EXTENDING SKILLS WITH TRIGONOMETRY

7 3. Solve for v, using graphing technolog. Answer to the nearest degree, 0 v 360. (a) 3 sin v sin v 0 (b) 8 cos v cos v 0 (c) 4 cos v 3 cos v 0 (d) 5 sin (v) 4 sin (v) 3 4. Solve for x using graphing technolog. Answer to the nearest hundredth, 0 x p. (a) 3 cos x 5 cos x (b) sin x 3 sin x 0 (c) 4 cos x 7 cos x (d) 3 sin x sin x 0 5. Check Your Understanding (a) Give an example of a quadratic trigonometric equation. (b) Can factoring be used to solve all quadratic trigonometric equations? Explain. (c) What two factors pla a role when determining the number of solutions of a quadratic trigonometric equation? C 6. Solve each equation without using graphing technolog. (a) 3 tan (x), 0 x 360 (b) sin v 3 cos v, 0 v p. (Hint: Square both sides of the equation and check for extraneous roots.) The Chapter Problem What Time Is It? Appl what ou have learned to answer these questions about the Chapter Problem on page 494. CP7. CP8. CP9. A 70-cm long pendulum is released at a distance of 8 cm from rest. (a) Determine the period and the time of one complete swing. (b) Graph the function that models this situation. Explain the meaning of positive and negative values of the dependent variable, x. (c) Determine the pendulum s distance from the rest position after s. (d) Determine four possible values for t when the pendulum is 6 cm from the centre point. Use the model x M cos t 98 0 l to determine an expression that represents the period of the function when l 70 cm. How long does the pendulum in a grandfather clock take to complete one full period? (A period is two complete swings.) CP0. Use our answers from CP8 and CP9 to create an equation to determine the length of the pendulum in a grandfather clock. Solve the equation. 6.6 SOLVING QUADRATIC TRIGONOMETRIC EQUATIONS 547