1. 6.3 / 6.4 - Radian Measure and the Unit Circle
Vocab:
Angle: Determined by rotating a ray about its endpoint.
Terminal Side
Vertex
Initial Side

2. Angle Coordinate System+  x
Standard Position  -

3. When arc length = radius, we have one radian
Definition of a Radian y 
arc length
= s x
radius = r
When arc length = radius, we have one radian
(When s = r)

4. Radian Measure
Radian value of an angle  =
Circumference of a circle = 2r
Radian measurement of a full circle =
Radians are unit-less (no symbol)

5. Radian Landmarks y 
0 and 2 x

6. Radian Landmarks y 
0 and 2 x

7. Other angles less than 90

8. Coterminal addition and subtraction
Ex: Find an angle that is coterminal to A

9. Complementary & Supplementary Angles
Complementary = 2 angles adding to…
900
Supplementary = 2 angles adding to…
1800

10. Weird Angles
ex:
Find complement and supplement…
For complement, subtract FROM
For supplement, subtract FROM

11. Degree / Radian Conversion
To convert radians into degrees, multiply by
ex:
To convert degrees into radians, multiply by
ex: 3000

12. Arc Length
An arc with central angle measure  radians has length:
where r is the radius length of the angle.
ex: The second hand on a clock is 6 in. long. How far does the
tip of the second hand move in 15 seconds?
15 seconds is 1/4th of a circle… 90o
90o = /2 radians

13. ex: Find the central angle measure (in radians) of an arc of
length 5cm on a circle of radius 3cm….
 = 5/3 radians
Linear and Angular Speed
Avg speed of a moving object =
Distance
Time
Arc Length
Time
= Linear Speed
= r * angular speed
Angle
Time
= Angular Speed

14. ex: A merry-go-round makes 8 revolutions per minute.
a) What is the angular speed of the MGR in radians per min?
b) How fast is a horse 12 feet from the center traveling?
c) How fast is a horse 4 feet from the center traveling?
8 revolutions = 2(8) = 16radians per min
Linear speed = r * angular speed --> r = 12,
angular speed = 16 radians/min (from a)
12(16) = 192 ft/min
c) r = 4, angular speed = 16 radians/min
4(16) = 64 ft/min

15. The Radian Number Line p / 4 p / 2 p 2p 2p p / 4 p / 2 p 2p p / 4
p / 4
p / 2 p 2p 2p
p / 4
p / 2 p 2p
p / 4
p / 2 p 2p
p / 4
p / 2 p

16. The Radian Number Line as a Circle
p / 2
p / 4 p
0 / 2p

17. t (-x, y) t (x, y) t (-x, -y) t (x, -y) The Unit Circle
A circle with radius “1” y
x2 + y2 = 1
(0, 1)
t (-x, y) t
(x, y) y y
(-1, 0)
(1, 0) x -x x -y -y
t (-x, -y)
t (x, -y)
(0, -1)

18. For any arc length ‘t’, having coordinates (x, y) on the Unit Circle:
The Six Trigonometric Functions
“Reciprocal Functions”
Sine (sin) Cosine (cos) Tangent (tan)
Cosecant (csc) Secant (sec) Cotangent (cot)
For any arc length ‘t’, having coordinates (x, y) on the Unit Circle: 1
sin t = y
csc t = y 1
cos t = x
sec t = x
t (x, y) y x
tan t =
cot t = x y

19. p 6 Example: Evaluate the six trigonometric functions at t =
t ( x , y ) 6 p 2 3 1 , y
sin = 6 p 2 1
sin t = y 2 3
cos = 6 p x
cos t = x x
tan t = y 3
tan = 6 p
Reciprocal Functions
csc = 6 p 2 3
2 3
sec = 6 p
cot = 6 p 3

20. IC 6.3-A - pg 441 #’s 1, 2, 6, 8, 9, 13, 16, 20, 21, 24, 25, 29, 32, 33, 37.
IC 6.3-B - pg 441 #’s: 40, 41, 44, 49, 53, 54, 65, 66, 78.

21. Periodic Functions
A function is periodic if there exists a positive real number such that: for all t in the domain of f.
The least number c for which f is periodic is called the ‘period’ of f.
ex: sin
sin
also = sin

22. ex:
In general:

23. Domain and Range of Sine and Cosine

24. Trig functions of any angle
(x, y) q r y x
Things to understand
r is always positive (sum of squares)
r cannot be equal to 0
sin & cos are defined for any angle 

25. ex: Let (-3, 4) be a point on the terminal side of 
ex: Let (-3, 4) be a point on the terminal side of . Find sin, cos and tan of .
sub-in x, y and r
Signs by Quadrant
sin +
All +
(backwards Z)
tan +
cos +

26. ex: Given
and
find sin  and sec …
Which variable in the tangent value is negative?… x? or y?…
Clue:
Where is the cosine function positive?
Cosine is positive in quadrant I (doesn’t work for us in this case…) and quadrant IV.
Q(IV) means that the y-value has to be the negative…
so…

27. ‘Quadrant’ Angles (0, 1) (use unit circle coordinates)  (-1, 0)
 (1, 0)
(0, -1) A

28. Reference Angle
The acute angle ’ formed by the terminal side of and the horizontal (x) axis.
ex:

29. Using reference angles to evaluate non-acute angles
ex: Evaluate 1 60 2
(3rd Quadrant)

30. ex: Evaluate 1 30 2
(2nd Quadrant)

31. ex: Evaluate 1 45
(2nd Quadrant)