1 Properties of Sine and Cosine Functions MATH 130 Lecture on 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 = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 0010sin 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. y x y = sin x

4 y x Example: y = 3 cos x Example: Sketch the graph of y = 3 cos x on the interval [– , 4  ]. Partition the interval [0, 2  ] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. maxx-intminx-intmax y = 3 cos x 22 0x (0, 3) (, 0) (, 3) (, –3)

5 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. y x y = cos x

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| If |a| > 1, the amplitude stretches the graph vertically. If 0 1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis. y x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x y = 2sin x y = sin x

7 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. For b  0, the period of y = a sin bx is. For b  0, the period of y = a cos bx is also. If 0 < b < 1, the graph of the function is stretched horizontally. If b > 1, the graph of the function is shrunk horizontally. y x period: 2 period: 4

8 y x y = cos (–x) Graph y = f(-x) Use basic trigonometric identities to graph y = f (–x) Example 1: Sketch the graph of y = sin (–x). Use the identity sin (–x) = – sin x The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. Example 2: Sketch the graph of y = cos (–x). Use the identity cos (–x) = – cos x The graph of y = cos (–x) is identical to the graph of y = cos x. y x y = sin x y = sin (–x) y = cos (–x)

9 y x 0 20 –2 0y = –2 sin 3x 0 x Example: y = 2 sin(-3x) Example: Sketch the graph of y = 2 sin (–3x). Rewrite the function in the form y = a sin bx with b > 0 amplitude: |a| = |–2| = 2 Calculate the five key points. (0, 0) (, 0) (, 2) (, -2) (, 0) Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x period: =

10 The graph of y = A sin (Bx – C) is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the cycle is shifted from x = 0 to x = C / B. The number C / B is called the phase shift. amplitude = | A| period = 2  / B. The graph of y = A sin (Bx – C) is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the cycle is shifted from x = 0 to x = C / B. The number C / B is called the phase shift. amplitude = | A| period = 2  / B. x y Amplitude: | A| Period: 2  /B y = A sin Bx Starting point: x = C/B The Graph of y = Asin(Bx - C)

11 Example Determine the amplitude, period, and phase shift of y = 2sin(3x-  ) Solution: Amplitude = |A| = 2 period = 2  /B = 2  /3 phase shift = C/B =  /3

12 Example cont. y = 2sin(3x-  )

13 A common mistake… a is not amplitude; is amplitude. a may be positive or negative; amplitude is always positive. The standard forms for sine and cosine functions are: where a,b,c and d are constants.

14 In the standard form: a controls amplitude b controls period c controls phase shift d controls vertical shift

15 Amplitude Period: 2π/b Phase Shift: c/b Vertical Shift