4.2 – Trigonometric Functions: The Unit
Circle
Learning Target: be able to evaluate
trigonometric functions using the unit
circle.
The Unit Circle – is a circle centered at
the origin with a radius of 1.
𝑥2 + 𝑦2 = 1
Memorize the Unit Circle Shown Below:
A Trick for memorizing the unit circle:
https://www.youtube.com/watch?v=LE6dmczMc68
http://www.youtube.com/watch?v=1CiX
AP8XaBg
The Six Trigonometric Functions:
Common Trig.
And their inverses
Values
Sine
Cosecant
Cosine
Secant
Tangent
Cotangent
Definitions of Trigonometric Functions:
𝑦
sin 𝑡 = 𝑦
cos 𝑡 = 𝑥
tan 𝑡 =
𝑥
𝑥
1
1
cot 𝑡 =
csc 𝑡 =
sec 𝑡 =
𝑦
𝑦
𝑥
EX: Evaluate the six trigonometric
functions at each real number.
a.) 𝑡 =
2𝜋
3
b.) 𝑡 =
4𝜋
3
c.) 𝑡 = 2𝜋
a.) 𝑡 =
2𝜋
corresponds to the point
3
−1 √3
( , ).
2 2
√3
sin 𝑡 = 𝑦 =
2
−1
cos 𝑡 = 𝑥 =
2
1
2
csc 𝑡 = =
𝑦 √3
2√3
=
3
1
2
sec 𝑡 = =
𝑥 −1
= −2
3⁄
√
𝑦
2
tan 𝑡 = =
𝑥 −1⁄
2
𝑥
1
cot 𝑡 = =
𝑦 −√3
√3
=−
3
√3 2
=
∙
2 −1
= −√3
4𝜋
b.) 𝑡 = corresponds to the point
3
1
√3
(− , − ).
2
2
−√3
sin 𝑡 = 𝑦 =
2
1
cos 𝑡 = 𝑥 = −
2
1 −2
csc 𝑡 = =
𝑦 √3
−2√3
=
3
1
sec 𝑡 = = −2
𝑥
𝑦
tan 𝑡 =
𝑥
𝑥
1
cot 𝑡 = =
𝑦 √3
√3
=
3
2
√3
=−
∙−
2
1
= √3
c.) 𝑡 = 2𝜋 corresponds to the point
(1,0).
1 1
csc 𝑡 = =
𝑦 0
= 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
1 1
cos 𝑡 = 𝑥 = 1
sec 𝑡 = = = 1
𝑥 1
𝑥 1
𝑦 0
cot 𝑡 = =
tan 𝑡 = = = 0
𝑦 0
𝑥 1
= 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
EX: Evaluate the six trigonometric
sin 𝑡 = 𝑦 = 0
functions of 𝑡 = −
2𝜋
3
.
Since t is negative, we must find its
positive coterminal angle.
2𝜋
2𝜋 6𝜋 4𝜋
𝑡=−
+ 2𝜋 = −
+
=
3
3
3
3
t now corresponds to the point
−1 −√3
(
2
,
2
). This problem is now identical
to part (b) in the previous example.
Facts about Sine and Cosine Curves:
- The domain of both sine and cosine
functions is all real numbers.
- The range of both sine and cosine is
between -1 and 1. We can graph the
curves to confirm this.
- Both curves are periodic (repetitive in
nature). Their period is 2𝜋.
- The cosine and secant functions are
even (symmetric to the y-axis).
- The cosecant, tangent, and cotangent
functions are odd (symmetric to the
origin).
EX: Find the following:
a.) cos
9𝜋
3
b.) sin (−
11𝜋
2
)
2
c.) If tan(𝑡) = , find tan(−𝑡)
3
a.) We must find a coterminal angle for
9𝜋
3
9𝜋
3
that is between 0 and 2𝜋. 𝜃 =
− 2𝜋 =
9𝜋
3
−
6𝜋
3
= 𝜋. Theta is
now equivalent to the point
1,0). cos 𝑡 = 𝑥 = −1.
(-
b.)
−11𝜋
2
−11𝜋
+ 2𝜋 =
2𝜋 =
−7𝜋
2
+
2
4𝜋
2
=
+
4𝜋
−3𝜋
2
2
=
−7𝜋
2
+
𝜋
+ 2𝜋 = .
2
This corresponds to the point (0,1).
Sin 𝑡 = 𝑦 = 1.
c.) Since tangent is an odd function, we
know that tan(-t)= - tan(t). Thus,
tan(-t)= - tan(t) = - 2/3.
EX: Use a calculator to evaluate:
a.) sin
5𝜋
7
b.) csc 2
*We must make sure that our calculator
is in radian mode!
a.)
b.) csc 2 =
1
sin 2

Upon completion of this lesson, you
should be able to:
1. evaluate trig functions using the
unit circle.
2. Identify the facts about sine and
cosine curves and explain their
differences.
For more information, visit
http://www.mathsisfun.com/geometry/unit-circle.html
HW Pg.299 3-51 3rds, 59, 63-66