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13.4 – The Sine Function

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The Sine Function Use the graph. Find the value(s) of each of the following. 1.the period 2.the domain 3.the amplitude 4.the range

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The Sine Function Solutions 1.the period: 2 units 2.the domain: all real numbers 3. the amplitude: = 1 unit 4. the range: –1 y 1, where y is a real number 2222 < – < –

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Graphing the Sine Function A Video Introduction

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Highlights of the Sine Function The sine function, matches the measure of an angle in standard position with the y-coordinate of a point on the unit circle. Within one cycle of the function the graph will “zero” by touching the x axis three times ( ); reach a minimum value of -1 at and a maximum value of 1 at.

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The Sine Function Use the graph of the sine function. a. What is the value of y= sin for = 180°? b. For what other value(s) of from 0° to 360° does the graph of sin have the same value as for = 180°? The value of the function at = 180° is 0. When y = 0, = 0° and 360°.

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The General Equation Suppose:

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The Sine Function Estimate each value from the graph. Check your estimate with a calculator. a. sin 3 sin 3 = 0.1411200081 Use a calculator to check your estimate. The sine function reaches its median value of 0 at 3.14. The value of the function at 3 is slightly more than 0, or about 0.1.

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The Sine Function (continued) b. sin 2 sin = 1Use a calculator to check your estimate. 2 The sine function reaches its maximum value of 1 at, so sin = 1. 22

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Graphing the Sine Function Sketch the graph of Steps: 1.Determine the amplitude. In this case a = 2. 2.Determine the period using the formula. This will be the outer boundary of your graph. Period = 3. Use five points equally spaced through one cycle to sketch a cosine curve. The five–point pattern is zero-max–zero–min–zero. Plot the points. 8

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Graphing the Sine Function Sketch the graph of Steps: 4. Make a smooth curve through the points to complete your graph. 8

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The Sine Function Use the graph of y = sin 6. a. How many cycles occur in this graph? How is the number of cycles related to the coefficient of in the equation? b. Find the period of y = sin 6. The graph shows 6 cycles. 2 ÷ 6 =Divide the domain of the graph by the number of cycles. 3 The number of cycles is equal to the coefficient of. The period of y = sin 6 is. 3

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The Sine Function This graph shows the graph of y = a sin for values of a = and a = 3. 3434 a. Find the amplitude of each sine curve. How does the value of a affect the amplitude? In each case, the amplitude of the curve is | a |. b. How would a negative value of a affect each graph? When a is negative, the graph is a reflection in the x-axis. The amplitude of y = 3 sin is 3. The amplitude of y = sin is 1, and the amplitude of y = sin is. 3434 3434

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Step 1: Choose scales for the y-axis and the x-axis that are about equal ( = 1 unit). On the x-axis, mark one period (4 units). Step 2: Mark equal spaces through one cycle by dividing the period into fourths. Step 3: Since the amplitude is 3, the maximum 3 and the minimum is –3. Plot the five points and sketch the curve. The Sine Function a. Sketch one cycle of a sine curve with amplitude 3 and period 4.

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The Sine Function (continued) b. Use the form y = a sin b. Write an equation with a > 0 for the sine curve in part a. The amplitude is 3, and a > 0, so a = 3. The period is 4, and 4 =, so b =. 2 b2 b 2 An equation for the function is y = 3 sin x. 2

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Divide the period into fourths. Using the values of the amplitude and period, plot the zero-max-zero-min-zero pattern. The Sine Function Sketch one cycle of y = sin 3. 5353 | a | =, so the amplitude is. 5353 5353 b = 3, so there are 3 cycles from 0 to 2. =, so the period is. 2 b2 b 2 3 2 3 Sketch the curve.

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The Sine Function Find the period of the following sine curve. Then write an equation for the curve. According to the graph, one cycle takes 3 units to complete, so the period is 3. To write the equation, first find b. period = Use the relationship between the period and b. 2 b2 b 3 = Substitute. 2 b2 b b = Multiply each side by. 2 3 b3b3 2.094Simplify. Use the form y =a sin b. An equation for the graph is y = 5 sin. 2 3

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