5 4 Equations and Graphs of Trigonometric Functions

5. 4 Equations and Graphs of Trigonometric Functions Math 30 -1 1

Determine the angle given the ratio. Determine the solution(s) for the trigonometric equation . We seek the angle (the value of x) for which the cosine gives the ratio Graphically . Reference Angle The reference angle for Quadrant IV x = 30° x = 150° Two solutions: 30° and 150°. Math 30 -1 2

Multiple Solutions When the domain of the function is restricted (0° ≤ x < 360°), the solutions to the trigonometric equation must occur within that given restriction. When the domain of the function is not restricted, often there are multiple solutions. The solutions repeat themselves in multiples of 360° from each original solution. The general solutions to the equation are x = 30° + 360°n or x = 150° + 360°n, n ϵ I Math 30 -1 3

Determine the solutions for the trigonometric equation for the interval Method 1: Solve Graphically Graph the related function The solutions are the xintercepts of the graph of the related function. (0, 150) (0, 30) (0, 210) (0, 330) Math 30 -1 The solutions for the interval 0° ≤ x < 360° are x = 30°, 150°, 210°, 330°. 4

Determine the solutions for the trigonometric equation for the interval 30° Method 2: Solve algebraically 210° Quadrant III x = 30° x = 150° 330° Quadrant III Math 30 -1 x = 30° x = 150° 5

Solving Trigonometric Equations 2 sin 2 x + 3 sin x + 1 = 0 (2 sin x + 1)(sin x + 1) = 0 2 sin x + 1 = 0 Reference Angle Math 30 -1 6

Interpreting Graphs to Find Solutions The diagram below shows the graphs of two trig functions y = 4 sin 2 x and y = 6 sinx + 2, defined for 0 ≤ x ≤ 2 p. Describe how you could use this graph to estimate the solution to the equation (4 sin 2 x) (6 sinx + 2) = 0 for 0 ≤ x ≤ 2 p. y = 6 sinx + 2 y= 4 sin 2 x The solutions will be the points where the graphs intersect the x-axis. Therefore, the solutions are: x = 0, 3. 14, 6. 28, 3. 48, and 5. 94 7

Interpreting Graphs to Find Solutions Describe how you could use this graph to estimate the solution to the equation 4 sin 2 x = 6 sinx + 2 for 0 ≤ x ≤ 2 p. y = 6 sinx + 2 y= 4 sin 2 x The solutions will be the points where the graphs intersect. Therefore, the solutions are: x = 3. 426 and 5. 999. 8

Interpreting Graphs to Find Solutions Use a graph to solve the equation Therefore, the solutions are: x = 0 and 2. 2789 Math 30 -1 9

Using Technology to Find Solutions Determine the lowest possible value of x, to the nearest tenth for which The smallest x-value is -2. 9. Math 30 -1 10

Assignment Page 275 1, 3, 4 a, c, 5 c, d, 6 a, c, 8, 12, 13, 14, 15, 19 Math 30 -1 11