13.4 – The Sine Function

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

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 < –

Graphing the Sine Function A Video Introduction

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 .

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°.

The General Equation Suppose:

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 3.14. The value of the function at 3 is slightly more than 0, or about 0.1. sin 3 = 0.1411200081   Use a calculator to check your estimate.

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

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

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

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 6 is . 3

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.

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.

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

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 2 . = , so the period is . 2 b 3

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