Circle

1. Republic of the Philippines
DEPARTMENT OF EDUCATION
Region I
Division of Ilocos Sur

2. CIRCULAR
FUNCTIONS

3. OBJECTIVES
 illustrate the six circular functions,
 find its exact values using reference
angles,
 determine its domain and range and
 graph the six circular functions and
identify their (a) amplitude, (b) period,
and (c) phase shift.

4. Have you ever played
the Super Mario game?
Have you ever observed
Mario glide so smoothly
over game obstacles?

5. SOH-CAH-TOA is a mnemonic device in trigonometry that you
must know. It stands for sine-opposite-hypotenuse (SOH), cosine-
adjacent-hypotenuse (CAH), tangent-opposite-adjacent (TOA).
Studying the right triangle below (Figure 1) and looking at the sides
with respect to the angle θ, y is the opposite side, x is the adjacent
side, and r is the hypotenuse. Then
Figure 1
cos 𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
sec 𝜃 =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑛𝑐𝑒𝑛𝑡
sin 𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
csc 𝜃 =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
tan 𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
cot 𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

6. Circular functions which is commonly known as the trigonometric
functions because the radian measures of the angles are calculated
by the length and coordinates of the terminal point of the arc on the
unit circle.
Circular functions are function whose domain and range
correspond to the measures of angles with respect to the
trigonometric functions. The basic circular functions are sine, cosine
and tangent and their reciprocal functions are cosecant, secant and
cotangent respectively.

7. The difference is that the domain of the trigonometric functions is the
set of angles in standard position while the range of the circular
functions is real numbers.
Figure 2
The angle is 45°. Since it is a unit
circle, the intercepted arc is
𝜋
4
.
In Figure 2, a unit circle and an angle
in standard position were drawn on
the rectangular coordinate plane
whose center and vertex are at the
origin

8. Figure 3
Notice that if a vertical line is
drawn from point P to the initial
side, a right triangle is formed.
(See Figure 3)
Line segment OP (radius) has
a length of 1 unit since circle O
is a unit circle. Considering the
central angle
𝜋
4
, x is the length of
the adjacent side and y is the
length of the opposite side.

9. Let 𝜃 be an angle in the standard position and P(x,y) be a point on
the terminal side of the angle. Then we have the six-circular function as
follows:
sin 𝜃 =
𝑦
𝑟
= 𝑦 csc 𝜃 =
1
sin 𝜃
=
1
𝑦
cos 𝜃 =
𝑥
𝑟
= 𝑥 sec 𝜃 =
1
cos 𝜃
=
1
𝑥
𝑡𝑎𝑛 𝜃 =
𝑦
𝑥
; 𝑥 ≠ 0 cot 𝜃 =
1
tan 𝜃
=
𝑥
𝑦
; 𝑦 ≠ 0

10. Therefore, if the
coordinates of point P on
the unit circle are known,
finding the values of the 6
circular functions would
not be difficult. Revisit the
unit circle and the
coordinates of point P on
the special angles
Figure 4. (etc.usf.edu)

11. ILLUSTRATIVE EXAMPLES
1. Find the exact values of sin
3𝜋
2
,
cos
3𝜋
2
, and tan
3𝜋
2
Solution:
Let P
3𝜋
2
be the point on the unit circle and on the
terminal side of the angle in the standard position with
measure
3𝜋
2
rad. Then P
3𝜋
2
= (0, -1), and so
sin
3𝜋
2
= −1, cos
3𝜋
2
= 0 ,
but tan
3𝜋
2
is undefined
1
0
.

12. ILLUSTRATIVE EXAMPLES
2. Find the values of cos 60°
and tan 60°
Solution:
We may also approach the problem
using the properties of 30° - 60° right
triangles. Review the figure on the right.
Thus, the coordinates of point P at
θ=60° are
1
2
,
3
2
.
Therefore, we get
cos 60° =
1
2
and tan 60° =
3
2
∙ 2 = 3

14. Find the values of the six circular functions of 𝜃 given that the terminal point is
P −
2
2
,
2
2
.
Solution:
Given: x = −
2
2
and y =
2
2
, Therefore,
sin 𝜃 = 𝑦 =
2
2
csc 𝜃 =
1
2
2
=
2
2
=
2
2
⋅
2
2
=
2 2
2
= 2
cos 𝜃 = 𝑥 = −
2
2
sec 𝜃 =
1
−
2
2
=−
2
2
= −
2
2
⋅
2
2
=
2 2
2
= − 2
𝑡𝑎𝑛 𝜃 =
𝑦
𝑥
=
2
2
−
2
2
=
2
2
⋅ −
2
2
= −1 cot 𝜃 =
1
−1
= −1

15. Give the six circular functions of the angle 𝜃 generated by an arc
whose length is
5𝜋
6
.
Thus the six circular functions are,
sin 𝜃 = 𝑦 =
1
2
csc 𝜃 =
1
1
2
= 2
cos 𝜃 = 𝑥 = −
3
2
sec 𝜃 =
1
−
3
2
=−
2
3
= −
2
3
⋅
3
3
= −
2 3
3
𝑡𝑎𝑛 𝜃 =
𝑦
𝑥
=
1
2
−
3
2
=
1
2
⋅ −
2
3
= −
1
3
⋅
3
3
= −
3
3
cot 𝜃 =
1
−
3
3
= −
3
3
⋅
3
3
= −
3 3
3
=− 3

16. ACTIVITY 1
Directions: Find all the
exact values of the six
circular functions at
𝑃
𝜋
6
.
Simplify and
rationalize (if needed)
your final answers.