1. 13-4 The Sine Function
Hubarth
Algebra II

2. The sine function, 𝑦= sin 𝜃, matches the measure 𝜃 of an angle in standard position with the
Y-coordinate of a point on the unit circle. This point is where the terminal side of the angle
Intersects the unit circle.

3. Ex. 1 Interpreting the Sine Function in Degrees
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°.

4. Ex. 2 Estimating Sine Values in Radians
𝜋 2 𝜋 2𝜋
3𝜋 2
Ex. 2 Estimating Sine Values in Radians
Estimate each value from the graph. Check your estimate with a calculator.
b. sin 2
a. sin 3
The sine function reaches its
median value of 0 at 𝜋≈ 3.14.
The value of the function at 3
is slightly more than 0, or about 0.1.
The sine function reaches its maximum value
of 1 at 𝜋 2 , so sin 𝜋 2 = 1.
sin = 1 2
sin 3 =   

5. Ex 3 Finding the Period of a Sine Curve
Use the graph of y = sin 6 , for 0 ≤ ≤ 2 .
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

6. Ex. 4 Finding the Amplitude of a Sine Curve
This graph shows the graph of y = a • sin𝜃 for values of a = 3 4 and a = 3.
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.

7. Ex. 5 Sketching a Graph
a. Sketch one cycle of a sine curve with amplitude 3 and period 4.
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.
Step 1: Choose scales for the y-axis and the x-axis that are about equal ( 𝜋= 1 unit). On the x-axis, mark one period (4 units).

8. The amplitude is 3, and a > 0, so a = 3.
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

9. Ex. 6 Graphing From a Function Rule
Sketch one cycle of y = 5 3 sin 3𝜃 .
| 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

10. Ex. 7 Real-World Connection
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.
period = Use the relationship between
the period and b. 2 b
3 = Substitute.
b = Multiply each side by . 3
Use the form y = a sin b . An equation for the graph is y = 5 sin 2 3