5.3-The Graphs of the Sine and Cosine Functions

Size: px

Start display at page:

Download "5.3-The Graphs of the Sine and Cosine Functions"

Percival Dixon
6 years ago
Views:

5.3-The Graphs of the Sine and Cosine Functions Objectives: 1. Graph the sine and cosine functions. 2. Determine the amplitude, period and phase shift of the sine and cosine functions. 3.

1 5.3-The Graphs of the Sine and Cosine Functions Objectives: 1. Graph the sine and cosine functions. 2. Determine the amplitude, period and phase shift of the sine and cosine functions. 3. Find equations of sine and cosine functions. 4. Sketch 1 cycle of sine and cosine functions. 5. Solve application problems using graphs. Overview: The graphs of trigonometric functions are important for understanding their use in modeling physical phenomena such as radio, sound, and light waves and the motion of a spring or pendulum. The Graph of the Sine Function: When graphing the sine function it is customary to write it in the form y = sin x where the independent variable x represents an angle and the dependent variable y represents the sine of the angle. As with any function, we can create a table of values for the function and the graph may be sketched using these values. Example: Create a table of values for the function points on the rectangular coordinate system. y = sin x over the interval [ 0,2π ], and then plot the Solution: It is most convenient to use quadrantal angles and angles which are multiples of 30, 45, or 60 degrees since we know the exact values of these angles.

2 Periodic Functions: The previous graph was plotted over the interval [ 0,2π ] however because the domain of y = sin x is all real numbers, this only represents a portion of the entire graph. If we continue to plot points in both directions we see that the graph repeats itself. A function that repeats its shape infinitely is called a periodic function. The graph of a periodic function such as y = sin x is called a sine wave, a sinusoidal wave, or sinusoid. The period of a periodic function is the smallest interval over which the function repeats its shape. The period of y = sin x is 2 π. The interval [ 0,2π ] is called the fundamental cycle of the function. The fundamental cycle is the portion of the function that we generally graph. Amplitude: The amplitude of a sine wave is a measure of the height of the wave. It is the absolute value of the difference between the maximum and minimum y-coordinates on the wave. From the graph, we can determine that the amplitude of the function y = sin x is 1. Example: Find the amplitude of the function y = 3sin x. Solution: In a previous lesson on transformations, we learned that when we multiply a function by a constant it stretches the graph vertically. Consequently, every point on the graph of the function y = sin x will be stretched vertically by a factor of three to create the graph of y = 3sin x. Therefore, the maximum value will be 3 rather than 1 and the minimum value will be -3 rather than -1. The amplitude is therefore: A = 1 (3 2 ( 3) = 3 In general, the amplitude of the function y = Asin x is A

The Graph of the Cosine Function: When graphing the cosine function it is customary to write it in the form y = cos x where the independent variable x represents an angle and the dependent variable y

3 Properties of the Sine Function: The table below summarizes the important properties of the sine function y = sin x. As with other functions we have studied, it is not necessary to memorize these properties but rather they can be determined by sketching the graph. The Graph of the Cosine Function: When graphing the cosine function it is customary to write it in the form y = cos x where the independent variable x represents an angle and the dependent variable y represents the cosine of the angle. As with any function, we can create a table of values for the function and the graph may be sketched using these values. Example: Create a table of values for the function points on the rectangular coordinate system. y = cos x over the interval [ 0,2π ], and then plot the Solution: As with the sine function, it is most convenient to use quadrantal angles and angles which are multiples of 30, 45, or 60 degrees since we know the exact values of these angles. The period of y = cos x is 2 π. The fundamental cycle of y = cos x is [ 0,2π ].

4 Periodic Functions: The previous graph was plotted over the interval [ 0,2π ] however because the domain of y = cos x is all real numbers, this only represents a portion of the entire graph. If we continue to plot points in both directions we see that the graph repeats itself. Amplitude: The amplitude of a cosine wave is a measure of the height of the wave. It is the absolute value of the difference between the maximum and minimum y-coordinates on the wave. From the graph, we can determine that the amplitude of the function y = cos x is 1. Example: Find the amplitude of the function y = 2cos x. Solution: In a previous lesson on transformations, we learned that when we multiply a function by a constant it stretches the graph vertically. Consequently, every point on the graph of the function y = cos x will be stretched vertically by a factor of 2 to create the graph of y = 2cos x. Therefore, the maximum value will be 2 rather than 1 and the minimum value will be -2 rather than -1. The amplitude is therefore: A = 1 (2 2 ( 2) = 2 In general, the amplitude of the function y = Acos x is A

5 Properties of the Cosine Function: The table below summarizes the important properties of the sine function y = cos x. As with other functions we have studied, it is not necessary to memorize these properties but rather they can be determined by sketching the graph. Transformations of Sine and Cosine: In a previous lesson we discussed how changes in a formula affect the graph of a function. We know the changes that cause horizontal or vertical translations, reflections, and stretching or shrinking. Horizontal Translation - Phase Shift: A phase shift of the graph y = sin( x C) or y = cos( x C) is C. If C > 0, then all points on the graph are shifted to the left C units. If C < 0, then all points on the graph are shifted to the right C units. π Example: Determine the phase shift of the function y = cos x, then graph the function. 2 π π Solution: The phase shift is and because > all points on the graph of y = cos x will be shifted horizontally 2 π units to the right.

6 Vertical Translation: The function y = sin x + D is a vertical shift of the graph y = sin x D units and the function y = cos x + D is a vertical shift of the function y = cos x D units. If D > 0, the shift is upward D units. If D < 0, the shift is downward D units. Example: Sketch the graph the function y = sin x + 2 Solution: From the graph of y = sin x, shift all points upward 2 units. Changing the Period: The period of a periodic function can be changed by replacing x by a multiple of x. The period P of the function y = sin( Bx) and y = cos(bx) for B > 0 is given by 2π P = B Example: Graph the function y = cos( 3x) and determine the period. Solution: The period is than every 2π units. 2π 2π P = =. This means that the function will repeat every B 3 2π units rather 3

7 Reflection over the X-Axis: The graph of the function y = sin x is a reflection over the x-axis of y sin x =. Example: Graph the function y = 2sin x Solution: This graph has an amplitude of 2 and is reflected over the x-axis. The General Sine Wave: We can use any combination of translating, reflecting, phase shifting, stretching or shrinking, or period changing in a single trigonometric function. The graph of y = Asin[ B( x C)] + D or y = Acos[ B( x C)] + D is a sine wave with an amplitude of A, period 2 π, phase shift C, and vertical translation D. B Procedure for Graphing a Sine Wave: Here is a recommended procedure for graphing the function 2π 1. Sketch 1 cycle over the interval 0, B. 2. Change the amplitude of the cycle according to the value of A. 3. If A < 0, reflect the curve in the x-axis. 4. Translate the cycle C units horizontally. 5. Translate the cycle D units vertically. y = Asin[ B( x C)] + D.

8 Example: Graph the function y = 2 + sin (2x - π) Solution: Example: Graph the function Solution: 1 π y = 2 + cos 4 x + 2 2

9 Example: The volume of air v in cubic centimeters in the lungs of a certain distance runner is modeled by the equation v = 400 sin(60π t) + 900, where t is time in minutes. What are the maximum and minimum volumes of air in the runner s lungs at any time? How many breaths does the runner take per minute? Solution: The equation v = 400 sin(60π t) has a vertical translation of 900 cubic centimeters upward. It also has amplitude of 400 cubic centimeters. Therefore the maximum volume of air in the runners lungs will be =1300, and the minimum volume will be =500. The number of breathes is represented by the frequency which is the reciprocal of the period. The period is determined by: 2π 1 P = = 60π 30 Therefore, the frequency is: The runner is taking 30 breathes per minute. F = 1 = Example: The velocity v of blood at a valve in the heart of a certain math student during an exam is modeled by the equation v = 4 cos(600t) + 8 where v is in centimeters per second and t is time in seconds. 1. What are the maximum and minimum velocities of the blood at this valve? 2. What is the student s heart rate during the exam? Express answer as an exact value. Solution: 1. The equation v = 4 cos(600t) + 8 has a vertical translation of 8 cm/sec. upward. It also has amplitude of 4 cm/sec. Therefore the maximum velocity will be 8+4=12, and the minimum velocity will be 8-4=4. 2. The number of breathes is represented by the frequency which is the reciprocal of the period. The period is determined by: 2π π P = = Therefore, the frequency is: F = = π π 300 The velocity of the blood through the valve during an exam is 300 cm/sec. π

10 Example: For the past three years the manager of a coffee shop has observed that revenue reaches a high of about $40,000 in December and a low of about $10,000 in June, and that the graph of the revenue looks like a sinusoid. If the months are numbered 1 through 36 with 1 corresponding to January, then what are the period, amplitude, phase shift, and vertical translation for this sinusoid? Write a formula for the curve and find the approximate revenue for April. Solution: We want to write the equation is the form of the General Sine Wave y = Asin[ B( x C)] + D The revenue is following a yearly pattern (repeating the cycle) so the period is 12 months. We may then find B as follows: 2π 2π π P = 12 = B = B B 6 Next, let s find the amplitude A and the vertical translation D. The difference between the high of $40,000 and the low of $10,000 is $30,000. Therefore: A = $30,000/2 = $15,000 D = $40,000-$15,000 = $25,000 π y = 15,000 sin[ ( x C)] + 25,000 6 To find the phase shift C, we will need to compare the given information to a normal sine curve (see chart below). A normal maximum value occurs at 90 degrees, however in this situation the maximum occurs in December. This may be thought of as a phase shits 3 units to the left or 270 degrees to the right. Because the shortest distance is to the left, the phase shift will be -3 units which is equivalent to 90 degrees. Therefore, in radians the phase shift is π 2 Max M M M Degrees # Month D J F M A M J J A S O N D The final formula is: π π y = 15,000sin[ ( x + )] + 25,

Section 5.2 Graphs of the Sine and Cosine Functions

Section 5.2 Graphs of the Sine and Cosine Functions We know from previously studying the periodicity of the trigonometric functions that the sine and cosine functions repeat themselves after 2 radians.