1. Chapter 14 Trigonometric Graphs, Identities, and Equations
Math III Accelerated
Chapter 14
Trigonometric Graphs, Identities, and Equations

2. Warm Up 14.1
Find the sine, cosine, and tangent for each.

3. 14.1 Graph Sine, Cosine, and Tangent Functions
Objective:
Graph sine, cosine, and tangent functions.

4. Periodic Function
A function is periodic if it repeats at regular intervals.
A cycle is the shortest repeating portion.
The period is the horizontal length of one cycle.
The amplitude is one-half the difference of the maximum, M and the minimum, m.

5. Examples of Periodic Functions

6. Sine Create a table of values for y = sin x & graph. x 0° 30° 45° 60°
90°
135°
180°
270°
360°
sin x

7. Graph of y = sin x
Domain: Range:
Period: Amplitude:
Max: Min:
Zeros:

8. Sine Function The function y = sin x is the parent function.
A simplified version of the general equation for sine is
where |a| = amplitude
b = number of cycles in one period
That is, period =

9. Graph y = sin x By Hand Use five key points in one period:
Zero, Max, Zero, Min, Zero

10. Example 1
Graph y = 2 sin x

11. Example 2
Graph y = sin 4x

12. Cosine
Create a table of values for y = cos x & graph. x π 2π
cos x

13. Graph of y = cos x
Domain: Range:
Period: Amplitude:
Max: Min:
Zeros:

14. Cosine Function The function y = cos x is the parent function.
A simplified version of the general equation for cosine is
where |a| = amplitude
b = number of cycles in one period
That is, period =

15. Graph y = cos x By Hand Use five key points in one period:
Max, Zero, Min, Zero, Max

16. Example 3
Graph

17. Example 4
Graph y = –3 cos x

18. Example 5 (NTG Ex. 2)
Write a sine function with an amplitude of 3 and a frequency of 1000.

19. Tangent Create a table of values for y = tan x & graph. x –1.57 –1.5
1.5
1.57
tan x

20. Graph of y = tan x
Domain:
Range:
Period:
Zeros:
VAs:

21. Tangent Function The function y = tan x is the parent function.
The simplified version of the general equation for tangent is
where b = number of cycles in one period
That is, period =

22. Graph y = tan x By Hand Find the VAs Solve for x: and .
Period is distance between VAs.
Find the zero
Midway between the VAs

23. Example 6 (NTG Ex. 3)
Graph one period of the function y = 2 tan x.

24. Example 7 (NTG CP 5)
Graph one period of the function y = tan 4x.

25. Example 8 (NTG CP 6)
Graph one period of the function y = tan πx.

26. Homework 14.1
Practice 14.1