Circular Functions

Circular Functions

2. A circular function can be defined in
terms of an arc length and the
coordinates of the terminal point of
the arc on the unit circle.

3. Let ?????? be an angle in the standard
position and let the point (x, y) be a
point on the terminal side of ??????.

4. ??????
NOTE: THE RADIUS OF UNIT
CIRCLE IS
EQUALS TO 1
Arc Length
Arc Length= 2????????????
??????
�????????????

Arc Length: s= r??????
Terminal Point
UNIT CIRCLE

5. Vertex
Terminal point
Terminal side
Initial side
Coordinates
(x,y) 3
2
,
1
2

??????

6. sin ?????? =
�
�
= y

cos ?????? =
�
�
= x

tan ?????? =
�
�
;x≠0

csc ?????? =
1
sin??????
=
1
�
; y ≠0

sec ?????? =
1
cos??????
=
1
�
; x ≠0

cot ?????? =
1
tan??????
=
�
�
; y ≠0

7. The sine, cosine, and tangent functions are
known as the basic circular function. The
cosecant, secant, and cotangent functions
are called the reciprocal function because
they are simply reciprocal of the basic
circular functions.

8. EXAMPLE:

Let P
3
2
,
1
2
be the terminal point of an arc
length s on the unit circle. Give the values
of six circular functions of s.

9. s
3
2
,
1
2

x,y
Let P
3
2
,
1
2
be the terminal
point of an arc length s on
the unit circle. Give the
values of six circular
functions of s.

10. SOLUTION:

The coordinates of the terminal point s
are x=
3
2
and y=
1
2
.

11. sin ?????? =
�
�
= y

cos ?????? =
�
�
= x

tan ?????? =
�
�
;x≠0

csc ?????? =
1
sin??????
=
1
�
; y ≠0

sec ?????? =
1
cos??????
=
1
�
; x ≠0

cot ?????? =
1
tan??????
=
�
�
; y ≠0

12. sin s =
1
2


cos ?????? =
3
2


tan ?????? =
1
2
3
2

=
1
2
×
2
3
=
1
3
×
3
3
=
3
3

13. csc ?????? =
1
1
2
=
1
1
×
2
1
= 2

sec ?????? =
1
3
2
=
1
1
×
2
3
=
2
3
×
3
3
=
23
3


cot ?????? =
1
3
3
=
1
1
×
3
3
=
3
3
×
3
3
=
33
3
= 3

14. EXAMPLE:
Give the values of six circular functions of s.

1. Let P −
2
2
,
2
2
be the terminal point of an arc
length s on the unit circle. Give the values of six
circular functions of s.

15. sin s =
2
2


cos ?????? =−
2
2


tan ?????? =
2
2
−
2
2
=
2
2
× −
2
2
= -1

16. csc ?????? =
1
2
2
=
1
1
×
2
2
=
2
2
×
2
2
=
22
2
= 2

sec ?????? =
1
−
2
2
=
1
1
× -
2
2
=-
2
2
×
2
2
= -
22
2
= −2

cot ?????? =
1
−1
=
1
1
×
1
−1
=
1
−1
= -1

17. EXAMPLE:
Give the values of six circular functions of s.

1. Let P
1
2
,
3
2
be the terminal point of an arc
length s on the unit circle. Give the values of six
circular functions of s.

18. sin s =
3
2


cos ?????? =
1
2


tan ?????? =
3
2
1
2
=
3
2
×
2
1
= 3

19. csc ?????? =
1
3
2
=
1
1
×
2
3
=
2
3
×
3
3
=
23
3


sec ?????? =
1
1
2
=
1
1
×
2
1
= 2

cot ?????? =
1
3
=
1
1
×
1
3
=
1
3
×
3
3
=
3
3

20. FINDING THE EXACT VALUES OF
TE CIRCULAR FUNCTIONS USING
REFERENCE ANGLE

21. Trigonometric functions are
sometimes called circular functions. This
is because the two
fundamental trigonometric functions –
the sine and the cosine – are defined as the
coordinates of a point P travelling around
on the unit circle.

22. 1
2
3
30-60-90 DEGREE TRIANGLE
Hypotenuse
????????????°
Given Angle
next

23. 45-45-90 DEGREE TRIANGLE

24. Reference angles can be used to
determine the values of the six circular
functions of a given angle ?????? as
determined by the terminal point of its
corresponding arc length.

25. UNIT CIRCLE
Quadrant 1 (+,+)
Quadrant 2 (-,+)

Quadrant 3 (-,-)

Quadrant 4(+,-)

26. REFERENCE
ANGLE= 30°????????????
??????
6

27. REFERENCE
ANGLE= 45° ????????????
??????
4

28. REFERENCE
ANGLE= 60° ????????????
??????
3

29. REFERENCE
ANGLE= 90° ????????????
??????
2

30. Q1= ??????
1
Q2= 180°- ??????
Q3= ??????−180°
Q4= 360°-??????
REFERENCE ANGLES:

31. QUADRANT 1
REFERENCE
ANGLE= 90°or
??????
2

Therefore, we use the special angles in the
first quadrant as reference to find the exact
values of the circular functions of the other
special angles in the second, third, and
fourth quadrants. We just need to
determine the quadrant where given
angle lies.

32. EXAMPLE:

Find the values of the six circular
functions of ?????? whose terminal side
is at
5??????
6
.
1

33. UNIT CIRCLE
Quadrant 1 (+,+)
Quadrant 2 (-,+)

Quadrant 3 (-,-)

Quadrant 4(+,-)

34. WE NEED TO FIND THE (x,y) coordinates.

sin= y
FIND THE sin(
5??????
6
)

cos= x
FIND THE cos(
5??????
6
)
SOLUTION:

35. RADIAN TO DEGREES:
5??????
6
×
180°
??????
= 150 °

36. Q1= ??????
1
Q2= 180°- ??????
Q3= ??????−180°
Q4= 360°-??????
REFERENCE ANGLES:

37. REFERENCE ANGLE:
180°- 150°= 30°

38. QUADRANT 2
REFERENCE
ANGLE= 30°????????????
??????
6

Quadrant 2 (-,+)

Q2= 180°- ??????

39. 60°
1
−3
2
30-60-90 DEGREE TRIANGLE
Quadrant 2 (-,+)
SOH CAH TOA
sin=
�������� ����
ℎ���������

cos=
??????��??????���� ����
ℎ���������

tan=
�������� ����
??????��??????���� ����

sin=
�������� ����
ℎ���������

sin=
1
2
y coordinate
cos=
??????��??????���� ����
ℎ���������

cos=
−3
2
x coordinate
−
3
2
,
1
2

40. UNIT CIRCLE
Quadrant 1 (+,+)
Quadrant 2 (-,+)

Quadrant 3 (-,-)

Quadrant 4(+,-)

41. 60°
1
−3
2
30-60-90 DEGREE TRIANGLE
Quadrant 2 (-,+)
SOH CAH TOA
sin=
�������� ����
ℎ���������

cos=
??????��??????���� ����
ℎ���������

tan=
�������� ����
??????��??????���� ����

tan=
�������� ����
??????��??????���� ����

tan=
1
−3
=
1
1
× -
1
3
= -
1
3
×
3
3
=
-
3
3

42. sin
5??????
6
=
1
2


cos
5??????
6
= -
3
2


tan
5??????
6
= -
3
3

43. csc
5??????
6
= 2

sec
5??????
6
=
1
−
3
2
=
1
1
×−
2
3
=-
2
3
×
3
3
=−
23
3


cot
5??????
6
=
1
−
3
3
=
1
1
×−
3
3
=-
3
3
×
3
3
=
33
3
= −3

44. EXAMPLE:
cos
5??????
4
=?

RADIAN TO DEGREES:
5??????
4
×
180°
??????
= 225 °

2

45. QUADRANT 2
REFERENCE
ANGLE= 45° ????????????
??????
4

Quadrant 2 (-,+)

cos
5??????
4
=?
cos
5??????
4
= -
2
2

Q2= 180°- ??????

46. EXAMPLE:
sin (
11??????
6
)=?

RADIAN TO DEGREES:
11??????
6
×
180°
??????
= 330 °
3

47. QUADRANT 4
REFERENCE
ANGLE= 30°????????????
??????
6

sin (
11??????
6
)=?
sin (
11??????
6
)= -
1
2

Quadrant 4 (+,-)

Q4= 3????????????°- ??????

48. EXAMPLE:
tan (-
2??????
3
)= tan
4??????
3


-
2??????
3
+2 ??????(1)=
4??????
3

RADIAN TO DEGREES:
4??????
3
×
180°
??????
= 240 °



4
Coterminal Angles

49. QUADRANT 3
REFERENCE
ANGLE= 60° ????????????
??????
3

Coterminal Angles
tan (-
2??????
3
)= tan
4??????
3
=?
tan
4??????
3
=?

Quadrant 3 (-,-)

Q3= ??????−180°

50. tan (-
2??????
3
)= tan
4??????
3
=?
tan
4??????
3
=?

tan ?????? =
�
�

x= -
1
2
; y= -
3
2

tan=
−
3
2
−
1
2
=−
3
2
× -
2
1
=
��
�
= �

52. Find the values of the six circular
functions of ?????? whose terminal side
is at
4??????
3
.

53. UNIT CIRCLE
Quadrant 1 (+,+)
Quadrant 2 (-,+)

Quadrant 3 (-,-)

Quadrant 4(+,-)

54. QUADRANT 3
REFERENCE
ANGLE= 60° ????????????
??????
3

Quadrant 3 (-,-)

Q3= ??????−180°

55. sin
4??????
3
=−
3
2


cos
4??????
3
= -
1
2


tan
4??????
3
= 3
csc
4??????
3
=−
23
3


sec
4??????
3
= 2

cot
4??????
3
=
3
3

ANSWERS:

56. UNIT CIRCLE
Quadrant 1 (+,+)
Quadrant 2 (-,+)

Quadrant 3 (-,-)

Quadrant 4(+,-)

57. Thank you!