3. Chapter 5
Trigonometric Functions of Real Numbers
5.1 The Unit Circle
John 3:16 For God so loved the world that he
gave his one and only Son, that whoever believes
in him shall not perish but have eternal life.
6. Assumptions
1. Much of this chapter is review for you
2. You will ask questions about concepts
you don’t understand or skills you
can’t do or remember
7. Assumptions
1. Much of this chapter is review for you
2. You will ask questions about concepts
you don’t understand or skills you
can’t do or remember
3. You have your Unit Circle with you in
in class and when doing homework
10. Two Approaches to Trigonometry
1. Unit Circle (Chapter 5)
2. Right Triangle (Chapter 6)
11. Two Approaches to Trigonometry
1. Unit Circle (Chapter 5)
2. Right Triangle (Chapter 6)
we work with both as each has its strengths
12. Two Approaches to Trigonometry
1. Unit Circle (Chapter 5)
2. Right Triangle (Chapter 6)
we work with both as each has its strengths
Note: In this chapter we will use mostly radians;
in Chapter 6 we will use mostly degrees
13. The Unit Circle is the circle of radius 1
centered at the origin
14. The Unit Circle is the circle of radius 1
centered at the origin
2 2
x + y =1
15. ⎛ 6 7 ⎞
Is the point ⎜ 6 , 6 ⎟ on the Unit Circle?
⎝ ⎠
16. ⎛ 6 7 ⎞
Is the point ⎜ 6 , 6 ⎟ on the Unit Circle?
⎝ ⎠
2 2
x + y =1
17. ⎛ 6 7 ⎞
Is the point ⎜ 6 , 6 ⎟ on the Unit Circle?
⎝ ⎠
2 2
x + y =1
2 2
⎛ 6 ⎞ ⎛ 7 ⎞
⎜ 6 ⎟ + ⎜ 6 ⎟ = 1
⎝ ⎠ ⎝ ⎠
18. ⎛ 6 7 ⎞
Is the point ⎜ 6 , 6 ⎟ on the Unit Circle?
⎝ ⎠
2 2
x + y =1
2 2
⎛ 6 ⎞ ⎛ 7 ⎞
⎜ 6 ⎟ + ⎜ 6 ⎟ = 1
⎝ ⎠ ⎝ ⎠
6 7
+ =1
36 36
19. ⎛ 6 7 ⎞
Is the point ⎜ 6 , 6 ⎟ on the Unit Circle?
⎝ ⎠
2 2
x + y =1
2 2
⎛ 6 ⎞ ⎛ 7 ⎞
⎜ 6 ⎟ + ⎜ 6 ⎟ = 1
⎝ ⎠ ⎝ ⎠
6 7
+ =1
36 36
13
≠1
36
No
20. ⎛ 35 ⎞
Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV.
⎝ 6 ⎠
Find y .
21. ⎛ 35 ⎞
Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV.
⎝ 6 ⎠
Find y .
2
⎛ 35 ⎞ 2
⎜ 6 ⎟ + y = 1
⎝ ⎠
22. ⎛ 35 ⎞
Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV.
⎝ 6 ⎠
Find y .
2
⎛ 35 ⎞ 2
⎜ 6 ⎟ + y = 1
⎝ ⎠
35 2 36
+y =
36 36
23. ⎛ 35 ⎞
Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV.
⎝ 6 ⎠
Find y .
2
⎛ 35 ⎞ 2
⎜ 6 ⎟ + y = 1
⎝ ⎠
35 2 36
+y =
36 36
2 1
y =
36
24. ⎛ 35 ⎞
Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV.
⎝ 6 ⎠
Find y .
2
⎛ 35 ⎞ 2
⎜ 6 ⎟ + y = 1
⎝ ⎠
35 2 36
+y =
36 36
2 1
y =
36
1
y=±
36
25. ⎛ 35 ⎞
Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV.
⎝ 6 ⎠
Find y .
2
⎛ 35 ⎞ 2
⎜ 6 ⎟ + y = 1
⎝ ⎠
35 2 36
+y =
36 36
2 1
y =
36
1
y=±
36
1
y=±
6
26. ⎛ 35 ⎞
Point P ⎜ , y⎟ is on the Unit Circle in quadrant IV.
⎝ 6 ⎠
Find y .
2
⎛ 35 ⎞ 2
⎜ 6 ⎟ + y = 1
⎝ ⎠ 1
We choose − as P is in Q IV
35 2 36 6
+y =
36 36 1
1 ∴ y=−
2
y = 6
36
1
y=±
36
1
y=±
6
27. When the initial ray rotates through some
angle, θ , its ending position is called
the terminal ray and the point of intersection
of the terminal ray and the Unit Circle is
the terminal point.
49. HW #1
Surround yourself with the best people you can find,
delegate authority, and don’t interfere as long as the
policy you’ve decided upon is being carried out.
Ronald Reagan