Download This problem is now identical to part (b) in the previous example.

10-4: The Unit-Circle Definition
of Cosine and Sine
Learning Target: I can identify
and use the definitions of sine
and cosine; use the properties of
a unit circle to find values of
trigonometric functions.
The Unit Circle – is a circle
centered at the origin with a
radius of 1.
𝑥2 + 𝑦2 = 1
Memorize the Unit Circle Shown
Below:
The Six Trigonometric
Functions:
Common Trig.
And their
Values
inverses
Sine
Cosecant
Cosine
Secant
Tangent
Cotangent
Definitions of Trigonometric
Functions:
sin 𝑡 = 𝑦
1
csc 𝑡 =
𝑦
cos 𝑡 = 𝑥 tan 𝑡 = 𝑦
𝑥
𝑥
1
cot 𝑡 =
sec 𝑡 =
𝑦
𝑥
EX: Evaluate the six
trigonometric functions at each
real number.
a.) 𝑡 = 120°
b.) 𝑡 = 240°
c.) 𝑡 = 360°
a.) corresponds to the point
−1 √3
( , ).
2 2
√3
sin 𝑡 = 𝑦 =
2
−1
cos 𝑡 = 𝑥 =
2
3⁄
√
𝑦
2
tan 𝑡 = =
𝑥 −1⁄
2
√3 2
=
∙
2 −1
= −√3
1
2
csc 𝑡 = =
𝑦 √3
2√3
=
3
1
2
sec 𝑡 = =
𝑥 −1
= −2
𝑥
1
cot 𝑡 = =
𝑦 −√3
√3
=−
3
b.) corresponds to the point
1
√3
(− , − ).
2
2
−√3
sin 𝑡 = 𝑦 =
2
1 −2
csc 𝑡 = =
𝑦 √3
−2√3
=
3
1
sec 𝑡 = = −2
𝑥
𝑥
1
cot 𝑡 = =
𝑦 √3
√3
=
3
1
cos 𝑡 = 𝑥 = −
2
𝑦
tan 𝑡 =
𝑥
2
√3
=−
∙−
2
1
= √3
c.) 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 functions of 𝑡 =
−120°.
sin 𝑡 = 𝑦 = 0
Since t is negative, we must find
its positive co-terminal angle.
𝑡 = −120 + 360 = 240°
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 360 𝑑𝑒𝑔𝑟𝑒𝑒𝑠.
- The cosine and secant
functions are even
(symmetric to the y-axis).
- The sine, cosecant, tangent,
and cotangent functions are
odd (symmetric to the
origin).
EX: Find the following:
a.) Cos 540 degrees
b.) sin −990°
2
c.) If tan(𝑡) = , find tan(−𝑡)
3
a.) We must find a coterminal
angle for 540 that is between
0 and 360. 𝜃 = 540 − 360 =
180. Theta is now equivalent
to the point
(-1,0).
cos 𝑡 = 𝑥 = −1.
b.) -990+360+360+360=90. 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 128.57°
b.) csc 2 degrees
*We must make sure that our
calculator is in DEGREE mode!
a.)
b.) csc 2 =
1
sin 2°
28.654
EX: What are the coordinates of
the image (0,1) under 𝑅50 ?
(cos 50° , sin 50°) = (0.643, 0.766)
EX: Find the sine of 270 degrees
with the unit circle.
-1
EX: Find the cosine of -90
degrees with the unit circle.
0
EX: Find the sine of -1170
degrees with both the unit circle
and a calculator.
-1
EX: Find the cosine of 810
degrees with both the unit circle
and a calculator.
0
http://www.youtube.com/watc
h?v=RLjyGKWMSx0
http://www.youtube.com/watc
h?v=Aqj3PFmsW54
http://www.youtube.com/watc
h?v=1CiXAP8XaBg
Upon completion of this lesson,
you should be able to:
1. recite the unit circle.
2. Identify the six trig
functions.
3. Explain the differences
between the sine and cosine
curves.
For more information, visit
http://www.mathsisfun.com/geometry/unitcircle.html
HW Pg.685 5-25, 28, 30
Quiz 10.1-10.4 tomorrow.