The document discusses the unit circle and trigonometric functions. It defines the unit circle as having a radius of 1 unit and center at the origin (0,0). The equation of the unit circle is provided as x2 + y2 = 1. Quadrantal angles are defined as angles whose terminal rays lie along one of the axes at 90°, 180°, 270°, and 360°. Trigonometric functions are defined in terms of the x- and y-coordinates on the unit circle. Special right triangles and their properties are also discussed.

1. Review: UNIT CIRCLE
The equation of a circle:
(x-h)2
+ (y-k)2
= r2
Where:
- (h, k) is the coordinate of the center of the
circle
- r is the radius
- (x, y) is the point of the circle

2. What is a UNIT CIRCLE?
A unit circle is a circle whose radius is equal
to 1 unit and its center is at the origin (0, 0).
Substituting the coordinates of the center
and the radius to the general equation of a
circle would determine the equation of a unit
circle.

3. Hence, the equation of the unit circle is
(x - 0)2
+ (y – 0)2
=1
or simply
x2
+ y2
= 1
The center (h, k) is (0, 0) and the radius (r) is
1.
Since x² = (−x)² for all x, and since the reflection of
any point on the unit circle about the x- or y-axis is
also on the unit circle, the above equation holds for
all points (x, y) on the unit circle, not only those in the
first quadrant.

4. Review: QUADRANTAL ANGLES
Quadrantal angles are angles in standard position
whose terminal ray lies along one of the axes.
Examples are: 90 ° (π/2 radian), 180 ° (π radian),
270 ° ( 3π/2 radian) and 360 ° (2π radian) and
their coterminal angles.

5. The Six Trigonometric functional
identities in a unit circle
Cos θ = x/r x
Sin θ = y/r y
Tan θ = y/x y/x
Sec θ = r/x 1/x
Csc θ = r/y 1/y
Cot θ = x/y x/y
x
y
r
θ

6. Circular Functions of Quadrantal Angles
Following the counterclockwise direction, the
quadrantal angles dividing the unit circle are as
follows: π/2 (90 °), π (180 °), 3(π)/2 (270 °), and
2π(360 °)If the direction point is clockwise, then the angles
become negative: -π/2 (-90), -π(-180), -3π/2 (-270), -
2π(-360).
(π) | 180 °
(π)/2 | 90 °
3(π)/2 | 270 °
2(π) | 360 °

7. As seen in this figure,
The coordinates of π/2 is (0, 1) and lies on Quadrant I;
the coordinates of π is (-1, 0) and lies on Quadrant II;
the coordinates of 3π/2 is (0, -1) and lies on Quadrant
III; while the coordinates of 2π is (0, -1) and lies on
Quadrant IV.
(π) | 180 °
(π)/2 | 90 °
3(π)/2 | 270 °
2(π) | 360 °

9. Radian Angle Coordinates Cos Sin Tan Sec Csc Cot
π/2 90 ( 0, 1) 0 1 1 0
π 180 (-1, 0) -1 0 0 -1
3π/2 270 (0, -1) 0 -1 -1 0
2π 360 (1, 0) 1 0 0 1
To summarize, this table presents the quadrantal
angles and their following coordinates, and
trigonometric values.
8
8
8
8
8
8
8
8

10. Oral Exercise
Find the value of the circular functions of the
given quadrantal angles.
1. sin π
2
2. sin 3π
2
3. sin 8π
4
1
-1
0
4. sin
8π
8
1=
=
=
5. cos 5π
2
6. sec 10
π2
0
-1
=
=
=

11. Now that the coordinates of the quadrantal angles are
defined, it is possible to identify the six trigonometric
functions of each angles.

12. Review on Special Triangles
Through the Pythagorean Theorem, the lengths of the sides
of 45° - 45° and 30° - 60° - 90°right triangles are derived.
x
y
r x2
+ y2
= r2

13. 7. cos
99
π99
8. sec 24
π2
-1
1
9. tan
π
2
10. cot 3π
2
11. tan 8π
4
12. cot 8π
8
8
0
0
8
Oral Exercise
Find the value of the circular functions of the
given quadrantal angles.
=
=
=
=
=
=

14. Review on Special Triangles
The length of the hypotenuse is
equivalent to the length of the
leg times square root of 2 in a
45 - 45 right triangle.
a
a
a√2
45°
45°
The length of the hypotenuse is
equivalent to twice the length of
the shorter leg (side opposite
30°), and the length of the
longer leg (side opposite 60°) is
equivalent to √3 times the
shorter leg.
a
a√3
2a
30°
60°

15. The trigonometric functions of special angles
would be determined with the aid of the unit
circle.
x
y
1
θ
Knowing the properties of these two special
triangles will allow you to easily find the
trigonometric functions of special angles, 30°,
45° and 60°.

16. Circular functions of multiples of
30° or π/6
x
y
1
30°

17. Cos θ = x
Sin θ = y
Tan θ = y/x
Sec θ = 1/x
Csc θ = 1/y
Cot θ = x/y
Circular functions of multiples of
30° or π/6
Cos
Sin
Tan
=
=
=
√3
2
1
2
√3
3
Sec
Csc
Cot
=
=
=
2√3
3
2
√3
Coordinates:

18. Circular functions of multiples of
150° or 5π/6

19. Circular functions of multiples of 210°
or 7π/6

20. Circular functions of multiples of
330° or 11π/6

21. To summarize, this table presents the circular
functions of π/3 and its multiples.

22. Circular functions of multiples of
45° or π/4
x
y
1
45°
4
π
4
3π
4
7π
4
5π
Since is in the 2nd
Quadrant, its coordinates are
Since is in the 3rd
Quadrant, its coordinates are
4
3π
4
5π
Since is in the 4th
Quadrant, its coordinates are
4
7π
),(
2
2
2
2
),(
2
2
2
2
−
),(
2
2
2
2
−−
),(
2
2
2
2
−

23. Cos θ = x
Sin θ = y
Tan θ = y/x
Sec θ = 1/x
Csc θ = 1/y
Cot θ = x/y
Circular functions of multiples of
45° or π/4
Cos
Sin
Tan
=
=
=
√2
2
√2
2
1
Sec
Csc
Cot
=
=
=
√2
Coordinates: ),(
2
2
2
2
4
π
4
π
4
π
4
π
4
π
4
π
1
√2

24. Cos
Sin
Tan
=
=
=
√2
2
√2
2
1
Sec
Csc
Cot
=
=
=
√2
Coordinates: )
2
2
,
2
2
(−
4
3π
1
√24
3π
4
3π
4
3π
4
3π
4
3π
-
-
Circular functions of
multiples of 135° or 3π/4
-
-

25. Cos
Sin
Tan
=
=
=
√2
2
√2
2
1
Sec
Csc
Cot
=
=
=
√2
Coordinates: )
2
2
,
2
2
( −−
4
3π
1
√24
3π
4
3π
4
3π
4
3π
4
3π
-
-
Circular functions of
multiples of 225° or 5π/4
-
-

26. Cos
Sin
Tan
=
=
=
√2
2
√2
2
1
Sec
Csc
Cot
=
=
=
√2
Coordinates: )
2
2
,
2
2
( −
4
3π
1
√24
3π
4
3π
4
3π
4
3π
4
3π
-
-
Circular functions of
multiples of 315° or 7π/4
-
-

28. Circular functions of multiples of
60° or π/3
x
y
1
60°
3
π
3
2π
3
5π
3
4π
Since is in the 2nd
Quadrant, its coordinates are
Since is in the 3rd
Quadrant, its coordinates are
Since is in the 4th
Quadrant, its coordinates are
),(
2
3
2
1
3
2π
3
4π
3
5π
),(
2
3
2
1
−
),(
2
3
2
1
−
),(
2
3
2
1
−−

29. Cos θ = x
Sin θ = y
Tan θ = y/x
Sec θ = 1/x
Csc θ = 1/y
Cot θ = x/y
Circular functions of multiples of
60° or π/3
Cos
Sin
Tan
=
=
=
1
2
Sec
Csc
Cot
=
=
=
2
Coordinates:
3
π
3
π
3
π
3
π
3
π
3
π
),(
2
3
2
1
√3
2
√3
2√3
3
√3
3

32. Exercise #1 – Part A
Directions: Write True if the statement is correct; otherwise,
changed the underlined word. Write your answers in a whole
sheet of paper. You only have 5 minutes to answer the following.
_________1. Quadrantal angles are angles whose terminal rays
lies in one of the axes.
_________2. (π)/2 lies in the positive x-axis.
_________3. Quadrantal real numbers are numbers whose
starting and terminal points lies on one of the axes.
_________4. The value of cos (π) is 0.
_________5. 3(π)/2 is equivalent to 360 ° .
Let us Check!
1. True
2. Negative
3. Arc lengths
4. -1
5. 2(π)

33. Exercise #1 – Part B
Directions: Identify the values being asked in the following. Write
your answers in a one whole sheet of paper. You only have 15
minutes to answer the following.
________1. sin(π)
________ 2. cot (2π)
________3. sec (5π/2)
________4. Csc (3 π/2)
________5. Tan (π/2)
Let us Check!
1. 0
2. Undefined
3. Undefined
4. -1
5. Undefined
________6. Cos (3π)
________7. tan (6π)
________8. Sin (11/2)
________9. Sec (7π/2)
________10. Cot (4π)
6. -1
7. 0
8. -1
9. -1
10. undefined

34. Exercises
1. sin45 + cot210 =
2. sec30+ tan 135=
3. csc630 – cot210 +
tan45=
4. sin240+ cos315=
5. sin90+ cos60=
Let Us Check!
1.
2.
3.
3
2
2
+
2
322 +
1
3
32
+
3
332 +
131 +−−
3−
4.
5.
2
2
2
3
+−
2
23 +−
2
1
1+
2
3

35. Exercises
1. sin60=
2. cot30=
3. tan150=
4. sec450=
5. csc120=
6. sin90=
1. sin60=
2. cot30=
3. tan150=
4. sec450=
5. csc120=
6. sin90=
Let Us Check!
1. sin60 =
2. cot30 =
3. tan150 =
4. sec450 =
5. csc120 =
6. sin90 =
√3
2
√3
√3
3
-
0
2√3
3
1