1. 13.4 – The Sine Function

2. I. Interpreting Sine Functions
The sine function, y = sin θ, matches the measure θ of an angle in standard position with the y – coordinate on the unit circle.
The periodic sine function has one cycle every 360° or 2π
Hence the sine of any degree greater than 2π, then the cycle is repeated.

3. The x-axis or domain will be thought of in terms of theta, θ
When graphing you have to pay attention to the domain of theta. Trig Cycles are repeated once every 360 degrees or 2π

4. The General Sine Curve:

5. Example 1: Graph, using a table the values of the sine curve in the domain
0 ≤ θ ≤ 2π. Use only the coordinates on the θ and y axes.

6. You can vary the period and amplitude of the sine curve to get different curves, either “extended” or more “frequent”
Done my multiplying the function by a constant, and theta by a constant

7. Y = 3 sin θ

8. Y = sin 2θ

9. II. Properties of the Sine Function
y = a sin bθ
ІaІ = the amplitude (the highest and lowest points on the curves.)
b = the number of cycles the curve makes from 0 to 2π
2π / b = the period of the cycle

10. Steps to Graphing y = a sin bθ
Step 1: draw your reference curve
Step 2: identify the domain and set up your graph.
Step 3: identify the amplitude and document
Step 4: identify the number of cycles in the domain that you have to graph
Step 4: identify the period you need to create one cycle
Step 5: Alter graph in needed in increments of 4 units on the theta axis to meet the criteria of the total cycles.

11. Example 2: how many cycles does the following have in the domain from 0 to 2π? Give the amplitude and period as well.
A) y = -5 sin 2θ
B) f(x) = .5 sin .5θ

12. Example 3: Graph the following. (check on the calculator)
A) y = -2 sin 2θ, for 0 < θ < 2π
B) y = 5 sin θ, for 0 < θ < 2π
C) y = ½ sin .5θ, for 0 < θ < 2π