1 Properties of Sine and Cosine Functions The Graphs of Trigonometric Functions

2 Properties of Sine and Cosine Functions 6. The cycle repeats itself indefinitely in both directions of the x-axis. Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 1. The domain is the set of real numbers. 5. Each function cycles through all the values of the range over an x-interval of. 2. The range is the set of y values such that.

3 Sine Function Graph of the Sine Function To sketch the graph of y = asinb x, assign values of x (angle measures in π radians) and y is the sine of your angle x. First locate the key points. These are the maximum points, the minimum points, and the intercepts. 0010sin x 0x Based on the graph, he maximum point is the value of a. The minimum point is the negative of a. y x y = sin x

4 Cosine Function Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1001cos x 0x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period (red portion of the graph). y x y = cos x

5 y x Example: y = 3 cos x Example: Sketch the graph of y = 3 cos x on the interval [0 , 2  ]. (Assign values of x (angle measures in π radians) and y is the sine of your angle x). maxx-intminx-intmax y = 3 cos x 22 0x (0, 3) (, 0) (, 3) (, –3)

6 Amplitude The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| y x y = sin x AMPLITUDE OF A TRIGONOMETRIC GRAPH The max. value of the graph is 1 and the min. value is -1. Therefore the amplitude is, a = |(max - min)/ 2| = |(1 - (-1))/ 2| a = 1

7 Amplitude The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically. y x y = 2sin x The max. value of the graph is 2 and the min. value is -2. Therefore the amplitude is, a = |(max - min)/ 2| = |(2 - (-2))/ 2| a = 2

8 y x y = sin x The max. value of the graph is 0.5 and the min. value is 0.5. Therefore the amplitude is, a = |(max - min)/ 2| = |(0.5 - (-0.5))/ 2| a = 0.5 If 0 1, the amplitude shrinks the graph vertically.

9 y x Period of a Function period: 2 The period of a function is the x interval needed for the function to complete one cycle. A cycle is length of one pattern. period of y = For b = 1, the period of y = a sin bx is. The period is 2π because the distance travelled by the curve to complete one cycle is 2π

10 If 0 < b < 1, the graph of the function is stretched horizontally. y x The period is 4π because the distance travelled by the curve to complete one cycle is 4π period: 4

11 If b > 1, the graph of the function is shrunk horizontally. y x The period is π because the distance travelled by the curve to complete one cycle is π

Determine the Amplitude and Period of the Trigonometric Function. 1.y = sin x Since in the function y = asin bx and y = acos bx, the amplitude is a and the period is 2π / |b|, then: Amplitude = 1 Period = 2π / |1| = 2π 12

2. y = cos 2x Amplitude = 1 Period = 2π / |2| = π 3. y = 3sin 4x Amplitude = 3 Period = 2π / |4| = π/2 13 Determine the Amplitude and Period of the Trigonometric Function.