5Highlights 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
6The 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°.
8The Sine FunctionEstimate each value from the graph. Check your estimate with a calculator.a. sin 3The 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.
9The Sine Function(continued)b. sin2The sine function reaches its maximum value of 1 at , so sin = 1.2sin = 1 Use a calculator to check your estimate.2
10Graphing the Sine Function Sketch the graph ofSteps: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 iszero-max–zero–min–zero.Plot the points.8
11Graphing the Sine Function Sketch the graph ofSteps:4. Make a smooth curve through the points to complete your graph.8
12The 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 ofb. Find the period of y = sin 6 .2 ÷ 6 =Divide the domain of the graph by the number of cycles.3The period of y = sin is .3
13The Sine FunctionThis graph shows the graph of y = a • sin for values of a = and a = 3.34a. 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 .34The 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.
14The Sine Functiona. 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.
15The 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 = .2bAn equation for the function is y = 3 sin x.2
16The Sine Function Sketch one cycle of y = sin 3 . 53| a | = , so the amplitude is .53Divide 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 is2b3
17The Sine FunctionFind 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.2b3 = Substitute.b = Multiply each side by .32.094 Simplify.Use the form y =a sin b . An equation for the graph is y = 5 sin23