1. Chapter 4
Pre-Calculus
OHHS

2. 4.3 The Circular Functions
Solve Trig Functions of Any Angle
Solve Trig Functions of Real Numbers
Understand Periodic Functions
Analyze the 16-point Unit Circle
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3. Anatomy of an Angle
Terminal Side
Vertex
Initial Side
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4. Angle Rotation Negative Angle Positive Angle Counter Clockwise
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5. Standard Position y
Terminal Side
Initial Side x
Vertex at (0,0)
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6. Quadrantal Angles
Terminal side of a standard position angle is on an axis.
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7. Coterminal Angles 45º+360ºn where n is an integer.
Angles with the same initial and terminal sides, but with different rotations.
All of these are coterminal angles.
45º
405º
-315º
765º
-675º
-1035º
1485º
-1395º
How did I find all these angles?
45º+360ºn
where n is an integer.
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8. Example
Find a positive angle and a negative angle that are coterminal with
(a) 30
30+360 = 390
(b)
30-360=-330
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9. Now You Try
P. 381, #1
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10. 1st Quadrant Trig sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = r y
P(x,y) r y θ x
csc θ =
sec θ =
cot θ =
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11. Example
Let θ be the acute angle in standard position whose terminal side contains the point (5, 3). Find the six trigonometric functions of θ.
P(5,3) 3 θ 5
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12. Now You Try
P. 381, #5
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13. Example
Let θ be the acute angle in standard position whose terminal side contains the point (-5, 3). Find the six trigonometric functions of θ.
P(-5,3) 3 θ -5
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14. Now You Try
P. 381, #11
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15. Reference Angle
The angle formed between the terminal side and the nearest part of the x-axis.
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16. Example
Find the exact values of the six trigonometric functions of 315
Reference
Angle
315
45 1
45 -1
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17. Now You Try
P. 381, #25
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18. Example Find the coordinates on the unit circle where θ = 210º x y 1
sin 210º =
cos 210º =
tan 210º =
sec 210º =
csc 210º =
cot 210º =
θ = 210º x
30º y 1
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19. Now You Try
P. 381, #29
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20. Trig Functions of Quadrantal Angles
cot(180) =
undefined
180
(-1, 0)
sin(180) =
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21. Now You Try
P. 381, #41
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22. Using One Trig Ratio to Find the Others
If sin θ = and tan θ < 0, find cos θ.
In which quadrant is this true? Q2 7 3 θ
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23. Using One Trig Ratio to Find the Others
If sec θ = 3 and sin θ > 0, find tan θ.
In which quadrant is this true? Q1 3 1
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24. Now You Try
P. 381, #43
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25. The Unit Circle 2 A circle with radius = 1 What is its circumference?
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26. The Unit Circle Wrapping Function
Circumference 2 
or 2  2
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27. Using the Unit Circle to Find Trig Ratios
The terminal side of any angle t intersects the unit circle at
(cos t, sin t)
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28. Trigonometric Functions on the Unit Circle
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29. Unit Circle Example
Find tan  
(-1, 0)
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30. Periodic Functions f(t + c) = f(t)
A function is periodic if there is a positive number c such that
f(t + c) = f(t)
for all values of t in the domain of f.
The smallest such number c is called the period of the function.
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31. Your Turn
Work Sheet 4.3
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32. Home Work
P. 381,
#2, 6, 8, 18, 20, 22, 26, 30, 36, 40, 44, 48, 50, 52,
61-66, 68, 70
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