1. Chapter 4 Trigonometric Functions
Pre-Calculus
Chapter 4
Trigonometric Functions

2. 4.5 Graphs of Sine and Cosine Functions
Objectives:
Sketch the graphs of basic sine and cosine functions.
Use amplitude and period to help sketch the graphs of sine and cosine functions.
Sketch translations of graphs of sine and cosine functions.
Use sine and cosine functions to model real-life data.

3. Sine and the Unit Circle
Applet:

4. Graph Sine
Complete the table of values and create a graph of the sine function. x
π/6
π/4
π/3
π/2
3π/4 π
3π/2 2π
sin x

5. Graph of Sine

6. Notes on the Graph of Sine
Domain:
Range:
Period:
Symmetry:
Five Key Points in One Period:

7. Graph Cosine
Complete the table of values and create a graph of the sine function. x
π/6
π/4
π/3
π/2
3π/4 π
3π/2 2π
cos x

8. Graph of Cosine

9. Notes on the Graph of Cosine
Domain:
Range:
Period:
Symmetry:
Five Key Points in One Period:

10. General Form of Sine & Cosine
The general form of sine and cosine are:
y = d + a sin (bx – c)
y = d + a cos (bx – c)
Where
|a| = amplitude
2π/b = period
c = horizontal shift
d = vertical shift

11. Example 1
On the same set of coordinate axes, sketch the graph of each function by hand on the interval [–π, 4π]. Compare these with the graph of the parent function.
y = ½ cos x
y = –3 cos x

12. Example 1

13. Example 2
On the same set of coordinate axes, sketch the graph of each function by hand on the interval [–π, 4π]. Compare these with the graph of the parent function.
y = sin (x/2)
y = sin 2x

14. Example 2

15. Graphs of Sine and Cosine
The graphs of y = d + a sin (bx – c) and y = d + a cos (bx – c) have the following characteristics:
Amplitude = |a|
Period = 2π/b
The left and right endpoints of a one-cycle interval can be determined by solving the equations bx – c = 0 and bx – c = 2π.

16. Example 3
Analyze and sketch the graph of
y = ½ sin (x – π/3)

17. Example 4
Analyze and sketch the graph of
y = –3 cos (2πx + 4π)

18. Example 5
Analyze and sketch the graph of
y = cos 2x

19. Example 6
Find the amplitude, period, and phase shift for the sine function whose graph is shown. Write an equation for this graph.

20. Example 7
Find the amplitude, period, and phase shift for the sine function whose graph is shown. Write an equation for this graph.

21. Mathematical Modeling
Sine and cosine functions can be used to model many real-life situations including:
Electric currents
Musical tones
Radio waves
Tides
Weather patterns

22. Example 8
Throughout the day, the depth of the water at the end of a dock in Bangor, WA varies with the tides. The table shows the depths (in feet) at various times during the morning.
Use a trig function to model this data.
A boat needs at least 10 feet of water to moor at the dock. During what times in the evening can it safely dock?
Time
Midnight
2 AM
4 AM
6 AM
8 AM
10 AM
Noon
Depth
3.1
7.8
11.3
10.9
6.6
1.7
0.9

23. Homework 4.5
Worksheet 4.5