1. MATHPOWER TM 12, WESTERN EDITION 5.1 5.1.1 Chapter 5 Trigonometric Equations

2. General Note about Graphing Calculator Windows The window variables of a graphing calculator can be expressed as: x: [Xmin, Xmax, Xscl] y: [Ymin, Ymax, Yscl] 5.1.2 Therefore, the window is: x: [0, 6.28, 1.57] y: [-4, 2, 1] For example:

3. 5.1.3 Using a Graph to Solve a Trigonometric Equation y = 3sin 4x From the graph of y = 3sin 4x, find all the solutions for 3sin 4x = 0, in the interval 0≤ x < 2  The solutions to the equation 3sin 4x = 0 are equivalent to finding the x-intercepts of the graph y = 3sin 4x. From the graph, the x-intercepts occur at Therefore, the solutions for 3sin 4x = 0 are

4. Using a Graph to Solve a Trigonometric Equation Find all the solutions to the equation 2cos x - 1 = 0. Method 1.Solve the equation for cos x and then graph both sides of the equation. The intersection of the two graphs is the solution. 0 ≤ x < 360 0 The solutions for 2cos x - 1 = 0 are 60 0 and 300 0. y = cos x 5.1.4

5. Method 2. Using a Graph to Solve a Trigonometric Equation [cont’d] Find all the solutions to the equation 2cos x - 1 = 0.0 ≤ x < 360 0 Graph y = 2cos x - 1. The x-intercepts are the solution. The solutions for 2cos x - 1 = 0 are 60 0 and 300 0. 5.1.5

6. Using a Graph to Find Solutions cos x = 0 cos 2x = 0 The graph of y = cos x has a period of 2 . Within the interval 0 ≤ x ≤ 2 , there are 2 zeros. y = cos 2x y = cos x The graph y = cos 2x has a period of . Therefore, there are two cycles in the interval 0 ≤ x ≤ 2  Each cycle has 2 zeros. 5.1.6 cos 3x = 0

7. Finding the General Solutions The graph of y = cos x has two x-intercepts in the interval 0 ≤ x ≤  Therefore, there are two solutions to the equation cos x = 0. If the restriction of the interval were removed, then the solutions would occur every 2  Therefore, the general solutions for y = cos x are: 5.1.7

8. Finding the General Solutions [cont’d] 5.1.8

9. The graph of y = cos 2x is related to the graph of y = cos x by a horizontal compression factor of 2. Setting 2x to the general solutions for cos x = 0 yields the following solutions: Finding the General Solutions 5.1.9

10. cos x = 0 cos 2x = 0 Solving Trig Equations 5.1.10

11. Suggested Questions: Pages 247 and 248 2, 3, 5, 6, 11, 12, 16-18, 21, 26 5.1.11