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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 2 < –

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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°? The value of the function at = 180° is 0. b. For what other value(s) of from 0° to 360° does the graph of sin have the same value as for = 180°? 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 The sine function reaches its median value of 0 at The value of the function at 3 is slightly more than 0, or about 0.1. sin 3 = Use a calculator to check your estimate.

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

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**Graphing the Sine Function**

Sketch the graph of Steps: Determine the amplitude. In this case a = 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? The graph shows 6 cycles. The number of cycles is equal to the coefficient of b. Find the period of y = sin 6 . 2 ÷ 6 = Divide the domain of the graph by the number of cycles. 3 The period of y = sin 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. 3 4 a. Find the amplitude of each sine curve. How does the value of a affect the amplitude? The amplitude of y = sin is 1, and the amplitude of y = • sin is . 3 4 The amplitude of y = 3 • sin is 3. 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.

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The Sine Function a. Sketch one cycle of a sine curve with amplitude 3 and period 4. 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 3: Since the amplitude is 3, the maximum 3 and the minimum is –3. Plot the five points and sketch the curve. Step 2: Mark equal spaces through one cycle by dividing the period into fourths.

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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 b An equation for the function is y = 3 sin x. 2

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**The Sine Function Sketch one cycle of y = sin 3 .**

5 3 | a | = , so the amplitude is . 5 3 Divide the period into fourths. Using the values of the amplitude and period, plot the zero-max-zero-min-zero pattern. Sketch the curve. b = 3, so there are 3 cycles from 0 to = , so the period is 2 b 3

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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 b 3 = Substitute. b = Multiply each side by . 3 2.094 Simplify. Use the form y =a sin b . An equation for the graph is y = 5 sin 2 3

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