Section 4.2 Notes Page 1
Ch4.2 Trigonometric Functions: The unit Circle.
Unit Circle is a circle centered at the origin with a radius of 1. This gives unit circle equation
of𝑥 2 + 𝑦 2 = 1. Here the letter t represents an angle measure. The point P=(x, y) represents a
point on the unit circle.
The following definitions are given based on this pictures.
1
sin 𝑡 = 𝑦 csc 𝑡 =
𝑦
1
cos 𝑡 = 𝑥 sec 𝑡 =
𝑥
𝑦
𝑥
tan 𝑡 =
cot 𝑡 =
𝑥
𝑦
5
12
Ex1. a) Suppose a point on the unit circle is (− 13 , − 13), find all six trigonometric values.
1
1
13
=
=−
𝑦 − 12
12
13
5
1
1
13
cos 𝑡 = 𝑥 = −
sec 𝑡 = =
=−
13
𝑥 − 5
5
13
12
5
𝑦 − 13 12
𝑥 − 13
5
tan 𝑡 = =
=
cot 𝑡 = =
=
𝑥 − 5
5
𝑦 − 12 12
13
13
sin 𝑡 = 𝑦 = −
12
13
csc 𝑡 =
√3 1
b) If 𝑃 = ( 2 , 2)
1 1 2
= = =2
𝑦 1 1
2
1
1
2
2√3
√3
cos 𝑡 = 𝑥 =
sec 𝑡 = =
=
=
2
𝑥 √3 √3
3
2
1
√3
𝑦
1
𝑥
√3
√3
2
tan 𝑡 = =
=
=
cot 𝑡 = = 2 =
= √3
1
𝑥 √3 √3
3
𝑦
1
2
2
sin 𝑡 = 𝑦 =
1
2
csc 𝑡 =
𝜋
c) Find the values of the trigonometric functions at 𝑡 = 2 .
Section 4.2 Notes Page 2
Domain and Range of Sine and Cosine
Domain is what values we can put into a trig function (t) and the range is the values it returns.
y sin t
x cos t
Domain: , 
Domain: , 
Range: [-1, 1]
Range: [-1, 1]
Section 4.2 Notes Page 3
Let’s look at the angle of t 45  or  . At this angle, we end up with the following triangle:
4

45 – 45 – 90 Triangle
45
1
We can use our definitions of sine, cosine, and tangent to find exact values:
2 2
sin 45 
2
2
45 
cos 45 
2
2
tan 45 
2 2
1
2 2
2 2
From our above definitions we can also find the following:
csc 45  1  2  2  2 2  2 ,
2
2
2 2
2


sec 45  1  2  2  2 2
2  2,
2
2 2
2

The following Even – Odd Properties will allow you to not deal with negative angles.
Even – Odd Properties
cos(t) cos t
sec(t) sec t
sin(t) sin t
csc(t)  csc t
tan(t) tan t
cot(t)  cot t
Periodic Properties
1
cos 45   1
1
Section 4.2 Notes Page 4
If we start at an angle and go around one revolution ( 360  or 2 radians) we will end up at the same angle we

started with. The k value is any integer, and represents how many revolutions are going around. If you want to
use degrees, replace the 2k in the equations below with 360k.
sin(t 2k ) sin t
csc(t 2k ) csc t
cos(t 2k ) cos t
sec(t 2k) sec t
tan(t 2k) tan t
cot(t 2k) cot t
Fundamental Identities
sin t  1

cs
ct

csc t 



sec t
tan t 
sec t  1

tan t 
1
cos t

cot t 
1
tan t
cot t
sin 2 t cos2 t 1
1 tan 2 t sec 2 t
sin t

cos t 
1
1
sin t
cos t
cot t 
cos t
sin t
1 cot 2 t csc 2 t
Section 4.2 Notes Page 5