1. Lesson 10: The Trigonometric Functions:
The Unit Circle Approach
Precalculus
Unit 3: Trigonometric Functions
2. At the end of the lesson, the learner is expected
to:
ο Use a unit circle to define trigonometric
functions of real numbers;
ο Recognize the domain and range of
trigonometric functions.
Learning Objectives:
Precalculus
Trigonometric Functions and the Unit Circle
4. 1. What do you notice about the figure in
the previous slide?
2. Can you name some of trigonometric
functions?
Answer the following key questions:
Precalculus
Trigonometric Functions and the Unit Circle
5. Trigonometric Functions using Unit Circle
sin π =
π¦
1
= π¦
cos π =
π₯
1
= π₯
tan π =
π¦
π₯
=
sin π
cos π
csc π =
1
π¦
=
1
sin π
sec π =
1
π₯
=
1
cos π
cot π =
π₯
π¦
=
cos π
sin π
Precalculus
Trigonometric Functions and the Unit Circle
8. Precalculus
Trigonometric Functions and the Unit Circle
Definitions
Amplitude- maximum ordinate (π¦) value.
Period- length of the smallest domain
interval which corresponds to a complete
cycle of values of a function.
15. Precalculus
Trigonometric Functions and the Unit Circle
Amplitude and Period
For the functions of the form:
π = π πππ ππ
π = π ππ¨π¬ ππ
Amplitude: π
Period: =
2π
π
17. Precalculus
Trigonometric Functions and the Unit Circle
π¦ =
sin π₯
cos π₯
; where cos π₯ β 0
Domain:
All values of y
Range:
One period
Tangent Function Graph
18. Precalculus
Trigonometric Functions and the Unit Circle
Amplitude and Period
For the function of the form: π = πππ§ ππ
Amplitude: None, since there are no
maximum value for π¦.
Period: =
π
π
20. Precalculus
Trigonometric Functions and the Unit Circle
π¦ = sec π₯ =
1
cos π₯
; where cos π₯ β 0
Domain:
All values of y
Range:
Secant Function Graph
21. Precalculus
Trigonometric Functions and the Unit Circle
π¦ = csc π₯ =
1
sin π₯
; where sin π₯ β 0
Domain:
All values of y
Range:
Cosecant Function Graph
22. Precalculus
Trigonometric Functions and the Unit Circle
π¦ = cππ‘ π₯ =
cos π₯
sin π₯
; where sin π₯ β 0
Domain:
All values of y
Range:
Cotangent Function Graph
23. Reference
οBacani, J. et. Al. Precalculus (Teaching Guide for Senior
High School). Published by Commission on Higher
Education, 2016.
οhttps://www.mathsisfun.com/