1. THE UNIT CIRCLE Precalculus Trigonometric Functions

2. Precalculus WU 10/14
Find one positive and one negative angle co-terminal with the given angle.

3. Objectives:
Find trig function values for special angles using the unit circle.
Evaluate Trig functions using the unit circle.
Use domain and period to evaluate trig functions.
Solve application problems using the unit circle.

4. Trigonometric Functions
The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle.
The sides of the right triangle are: θ
hyp
 the side opposite the acute angle ,
opp
 the side adjacent to the acute angle ,
adj
 and the hypotenuse of the right triangle.
The trigonometric functions are
sine, cosine, tangent, cotangent, secant, and cosecant.
sin  = cos  = tan  =
csc  = sec  = cot  =
opp
hyp
adj
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Trigonometric Functions

5. Definition: Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y), a point on the terminal side of  and r = y x r
(x, y) 
Definition: Trigonometric Functions of Any Angle

6. Example: Trigonometric Functions of Any Angle
Determine the exact values of the six trigonometric functions of the angle . y x
(3, 6) 
Example: Trigonometric Functions of Any Angle

7. So, we know that trigonometric function values are side length relationships of right triangles.
We can easily evaluate the exact values of trigonometric functions for special angles.

8. Geometry of the 30-60-90 Triangle2
Consider an equilateral triangle with each side of length 2.
60○
30○
30○
The three sides are equal, so the angles are equal; each is 60.
The perpendicular bisector of the base bisects the opposite angle. 1 1
Use the Pythagorean Theorem to find the length of the altitude,
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Geometry of the Triangle

9. Special right triangle relationships

10. Now, let’s apply it to the unit circle…
What does “unit circle” really mean?
It’s a circle with a radius of 1 unit.
What is the equation of the “unit circle”?

12. Let’s begin with an easy family…
What are the coordinates?
Now, reflect the triangle to the second quadrant…

13. What are the coordinates?
Now, reflect the triangle to the third quadrant…

14. What are the coordinates?
Now, reflect the triangle to the fourth quadrant…

15. What are the coordinates?

16. Now, reflect the triangle to the second quadrant.
Complete the family…
Now, reflect the triangle to the second quadrant.

17. Now, reflect the triangle to the third quadrant.

18. Now, reflect the triangle to the fourth quadrant.
What are the coordinates?
Now, reflect the triangle to the fourth quadrant.

19. What are the coordinates?

20. Let’s look at another “family”
Now, reflect the triangle to the second quadrant

21. What are the coordinates?
Now, reflect the triangle to the third quadrant

22. What are the coordinates?
Now, reflect the triangle to the fourth quadrant

23. What are the coordinates?

24. Trigonometric Values of Common Angles
Ordered pairs of special angles around the Unit Circle x y
(0, 1)
90°
120°
60°
Since r = 1…
135°
45°
30°
150°
(–1, 0)
180°
(1, 0) 0°
360°
210°
330°
315°
225°
240°
300°
(0, –1)
270°
Trigonometric Values of Common Angles

25. Important point: Since r = 1…
Because ordered pairs around the unit circle (x, y) represent sine and cosine, and the equation of the circle is ,
We have the following identity:

26. What if the radius is not 1?6 1
Trigonometric values are functions of the angle – ratios of sides of similar triangles remain the same. So it always holds that

27. Trigonometric Values of Common Angles
Trigonometric Values of Special Angles x y
(0, 1)
90°
120°
60°
135°
45°
30°
150°
(–1, 0)
180°
(1, 0) 0°
360°
210°
330°
315°
225°
240°
300°
(0, –1)
270°
Trigonometric Values of Common Angles

28. Domain and Range of Sine and Cosine
The domain of the sine and cosine function is the set of all real numbers.
(1, 0)
(–1, 0)
(0, –1)
(0, 1) x y
Unit Circle
Range
The point (x, y) is on the unit circle, therefore the range of the sine and cosine function is between – 1 and 1 inclusive.
Domain and Range of Sine and Cosine

29. Definition: Periodic Functions
A function f is periodic if there is a positive real number c such that
f (t + c) = f (t)
for all t in the domain of f. The least number c for which f is periodic is called the period of f. x y
Unit Circle
Periodic Function
t = 0, 2, …
Period
Definition: Periodic Functions

30. Example: Periodic Functions
Evaluate sin 5 using its period.
5 - 2 - 2 = 
sin 5 = sin  = 0 x y
(–1, 0)
Adding 2 to each value of t in the interval [0, 2] completes another revolution around the unit circle.
Example: Periodic Functions

31. You Try:
Evaluate sin
Evaluate cos

32. Can you?
Evaluate each of the following. Exact values only please.

33. Even and Odd Trig Functions
(1, 0)
(–1, 0)
(0,–1)
(0,1) x y
Remember: if f(-t) = f(t) the function is even if f(-t) = - f(t) the function is odd The cosine and secant functions are EVEN. cos(-t)=cos t sec(-t)=sec t The sine, cosecant, tangent, and cotangent functions are ODD. sin(-t)= -sin t csc(-t)= -csc t tan(-t)= -tan t cot(-t)= -cot t

34. Evaluating Trigonometric Functions
Example:
Evaluate the six trigonometric functions at  = .
(1, 0)
(–1, 0)
(0, –1)
(0, 1) x y 
Evaluating Trigonometric Functions

35. Evaluate the six trigonometric functions at
Example:
Evaluate the six trigonometric functions at
y =
x =

36. Example continued: -2

37. Evaluate the six trigonometric functions at
You Try:
Evaluate the six trigonometric functions at
(0, -1)
= -1
= y
= -1
= x
= 0
DNE
= 0
DNE
Does Not Exist

38. So, you think you got it now?
You Try:
Evaluate the six trigonometric functions at -1
Sin
Cos
Tan
Csc
Sec
Cot -1
So, you think you got it now?

39. Application:
A ladder 20 feet long leans against the side of a house.
The angle of elevation of the ladder is 60 degrees.
Find the height from the top of the ladder to the ground.

40. Application:
An airplane flies at an altitude of 6 miles toward a point
directly over an observer. If the angle of elevation
from the observer to the plane is 45 degrees, find the
horizontal distance between the observer and the plane. .

41. Homework
4.2 pg odd

42. Trig Races

43. HWQ 10/15
Evaluate each of the following. Exact values only please.