1. Circular
Trigonometric
Functions

2. Circular Trigonometric FunctionsY
circle…center at (0,0)
radius r…vector with
length/direction r θ X
angle θ… determines
direction

3. Quadrant II Quadrant I 360º Quadrant III Quadrant IV Y-axis 90º r r θ
Terminal side r r θ 0º
X-axis
180º
Initial side
360º
Quadrant III
Quadrant IV
270º

4. Quadrant II Quadrant I Quadrant III Quadrant IV Y-axis -270º -360º
X-axis
-180º
Terminal
side
Initial side 0º r θ
Quadrant III
Quadrant IV
-90º

5. angle θ…measured from positive x-axis,
or initial side, to terminal side
counterclockwise: positive direction
clockwise: negative direction
four quadrants…numbered I, II, III, IV counterclockwise

6. six trigonometric functions for angle θ
whose terminal side passes thru point
(x, y) on circle of radius r
sin θ = y / r csc θ = r / y
cos θ = x / r sec θ = r / x
tan θ = y / x cot θ = x / y
These apply to any angle in any quadrant.

7. For any angle in any quadrant
x2 + y2 = r2 …
So, r is positive by Pythagorean theorem.
(x,y) r y θ x

8. NOTE: right-triangle definitions are special case of circular
functions
when θ is in
quadrant I Y
(x,y) r y θ X x

9. *Reciprocal Identities
sin θ = y / r and csc θ = r / y
cos θ = x / r and sec θ = r / x
tan θ = y / x and cot θ = x / y

10. *Both sets of identities are useful to determine trigonometric
*Ratio Identities
*Both sets of identities are useful
to determine trigonometric
functions of any angle.

11. Students Take Classes Positive trig values in each quadrant: AllY
Students
All
all six positive
sin positive
(csc)
(-, +)
(+, +) II I X
III IV
Take
Classes
(-, -)
(+, -)
tan positive
(cot)
cos positive
(sec)

12. In the ordered pair (x, y), x represents cosine and
REMEMBER:
In the ordered pair (x, y),
x represents cosine and
y represents sine. Y
(-, +)
(+, +) II I X
III IV
(-, -)
(+, -)

13. Examples

14. #1 Draw each angle whose terminal side
passes through the given point, and find
all trigonometric functions of each angle.
θ1: (4, 3)
θ2: (- 4, 3)
θ3: (- 4, -3)
θ4: (4, -3)
SOLUTION

15. x = y = I r = (4,3) θ1 sin θ = cos θ = tan θ = csc θ = sec θ = cot θ =
SOLUTION

16. x = II y = r = (-4,3) θ2 sin θ = cos θ = tan θ = csc θ = sec θ =
cot θ =
SOLUTION

17. x = y = r = θ3 (-4,-3) III sin θ = cos θ = tan θ = csc θ = sec θ =
cot θ = θ3
(-4,-3)
III
SOLUTION

18. x = y = r = θ4 (4,-3) IV sin θ = cos θ = tan θ = csc θ = sec θ =
cot θ = θ4
(4,-3) IV
SOLUTION

19. Perpendicular II I line from point on circle always drawn
to the x-axis
forming a
reference triangle II I
ref θ2 θ1 X
ref θ3
ref θ4
III IV

20. is equal to trig function of its reference angle, or it differs
Value of trig
function
of angle in
any quadrant
is equal to trig function of its
reference angle,
or it differs
only in sign. Y II I
ref θ2 θ1 X
ref θ3
ref θ4
III IV

21. #2 Given: tan θ = -1 and cos θ is positive:
Draw θ. Show the values for x, y, and r.
SOLUTION

22. Given: tan θ = -1 and cos θ is positive:
Find the six trigonometric functions of θ.
SOLUTION

23. Calculator
Exercise

24. (First determine the reference angle.)
# 1 Find the value of sin 110º.
(First determine the reference angle.)
SOLUTION

25. (First determine the reference angle.)
#2 Find the value of tan 315º.
(First determine the reference angle.)
SOLUTION

26. (First determine the reference angle.)
#3 Find the value of cos 230º.
(First determine the reference angle.)
SOLUTION

27. Practice

28. #1 Draw the angle whose terminal side passes through the given point .
SOLUTION

29. Find all trigonometric functions for angle whose terminal side passes thru .
SOLUTION

30. #2 Draw angle: sin θ = 0.6, cos θ is negative.
SOLUTION

31. Find all six trigonometric functions: sin θ = 0.6, cos θ is negative.
SOLUTION

32. #3 Find remaining trigonometric functions:
sin θ = , tan θ = 1.000
SOLUTION

33. Find remaining trigonometric functions:
sin θ = , tan θ = 1.000
SOLUTION

34. Calculator
Practice

35. #1 Express as a function of a reference
#1 Express as a function of a reference angle and find the value: cot 306º .
SOLUTION

36. #2 Express as a function of a reference
#2 Express as a function of a reference angle and find the value: sec (-153º) .
SOLUTION

37. #3 Find each value on your calculator. (Key in exact angle measure.)
sin 260.5º
tan 150º 10’
SOLUTION

38. cot (-240º)
csc 450º
SOLUTION

39. cos 5.41
sec (7/4)
SOLUTION

40. π/2 = 1.57
2π = 6.28
π = 3.14
3π/2 = 4.71

41. Application

42. # 1 The refraction of a certain prism is
Calculate the value of n.
SOLUTION

43. #2 A force vector F has components Fx = - 4.5 lb and Fy = 8.5 lb.
Find sin θ and cos θ.
Fy = 8.5 lb θ
Fx=-4.5 lb
SOLUTION

44. Fy = 8.5 lb θ
Fx=-4.5 lb
SOLUTION