1. Precalculus Essentials
Fifth Edition
Chapter 4
Trigonometric Functions
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2. 4.2 Trigonometric Functions: The Unit Circle

3. Learning Objectives
Use a unit circle to define trigonometric functions of real numbers.
Recognize the domain and range of sine and cosine functions.
Find exact values of the trigonometric functions at
Use even and odd trigonometric functions.
Recognize and use fundamental identities.
Use periodic properties.
Evaluate trigonometric functions with a calculator.

4. The Unit Circle (1 of 2)
A unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this circle is

5. The Unit Circle (2 of 2)
In a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc. Both are given by the same real number t.

6. The Six Trigonometric Functions
Name
Abbreviation
sine
sin
cosecant
csc
cosine
cos
secant
sec
tangent
tan
cotangent
cot

7. Definitions of the Trigonometric Functions in Terms of a Unit Circle
If t is a real number and P = (x, y) is a point on the unit circle that corresponds to t, then

8. Example 1: Finding Values of the Trigonometric Functions (1 of 2)
Use the figure to find the values of the trigonometric functions at t.

9. Example 1: Finding Values of the Trigonometric Functions (2 of 2)
Use the figure to find the values of the trigonometric functions at t.

10. The Domain and Range of the Sine and Cosine Functions
The domain of the sine function and the cosine function is
the set of all real numbers.
The range of these functions is [−1,1], the set of all real numbers from −1 to 1, inclusive.

11. Exact Values of the Trigonometric Functions at t Equals Pi over 4 (1 of 3)
occur frequently. We use
the unit circle to find values of the trigonometric functions at
We see that point P = (a, b) lies on the line y = x. Thus, point P has equal x- and y-coordinates: a = b.

12. Exact Values of the Trigonometric Functions at t Equals Pi over 4 (2 of 3)
We find these coordinates as follows:
This is the equation of the unit circle.
Point P = (a, b) lies on the unit circle. Thus its coordinates satisfy the circle's equation.
Because a = b, substitute a for b in the previous equation.
Combine like terms.
Divide both sides of the equation by 2.
Because a > 0, take the positive square root of both sides.

13. Exact Values of the Trigonometric Functions at t Equals Pi over 4 (3 of 3)
We have used the unit circle to find the coordinates of point P = (a, b) that correspond to

14. Example 2: Finding Values of the Trigonometric Functions at t Equals Pi over 4 (1 of 2)

15. Example 2: Finding Values of the Trigonometric Functions at t Equals Pi over 4 (2 of 2)

16. Trigonometric Functions at Pi over 4

17. Even and Odd Trigonometric Functions
The cosine and secant function are even.
The sine, cosecant, tangent, cotangent function are odd.

18. Example 3: Using Even and Odd Functions to Find Values of Trigonometric Functions
Find the value of each trigonometric function:

19. Fundamental Identities
Reciprocal Identities
Quotient Identities

20. Example 4: Using Quotient and Reciprocal Identities (1 of 2)
Given
and
find the value of each of
the four remaining trigonometric functions.

21. Example 4: Using Quotient and Reciprocal Identities (2 of 2)
Given
and
find the value of each of
the four remaining trigonometric functions.

22. The Pythagorean Identities

23. Example 5: Using a Pythagorean Identity
Given
and
find the value of cos t
using a trigonometric identity.

24. Definition of a Periodic Function
A function f is periodic if there exists a positive number p such that f (t + p) = f(t)
For all t in the domain of f. The smallest positive number p for which f is periodic is called the period of f.

25. Periodic Properties of the Sine and Cosine Functions
The sine and cosine functions are periodic functions and have period

26. Periodic Properties of the Tangent and Cotangent Functions
The tangent and cotangent functions are periodic functions and have period

27. Example 6: Using Periodic Properties
Find the value of each trigonometric function:

28. Repetitive Behavior of the Sine, Cosine, and Tangent Functions
For any integer n and real number t,

29. Using a Calculator to Evaluate Trigonometric Functions
To evaluate trigonometric functions, we will use the keys on a calculator that are marked Sin, Cos, and Tan. Be sure to set the mode to degrees or radians, depending on the function that you are evaluating. You may consult the manual for your calculator for specific directions for evaluating trigonometric functions.

30. Example 7: Evaluating Trigonometric Functions with a Calculator
Use a calculator to find the value to four decimal places: