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Precalculus Trigonometric FunctionsThe Unit Circle 2015
Precalculus WU 10/14
Find one positive and one negative angle co-terminal with the given angle.
5
12
Objectives:
•Find trig function values for special angles using the unit circle.
•Evaluate Trig functions using the unit circle.
•Use domain and period to evaluate trig functions.
•Solve application problems using the unit circle.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
4
The six trigonometric functions of a right triangle, with an
acute angle , are defined by ratios of two sides of the triangle.
The sides of the right triangle are:
the side opposite the acute angle ,
the side adjacent to the acute angle , and the hypotenuse of the right triangle.
The trigonometric functions are
sine, cosine, tangent, cotangent, secant, and cosecant.
opp
adj
hyp
θ
sin = cos = tan =
csc = sec = cot =
opphyp
adj
hyp
hypadj
adj
opp
oppadj
opp
hyp
5
Trigonometric Functions
sin yr
cos xr
tan , 0y xx
Let be an angle in standard position with (x, y), a point on
the terminal side of and r = 2 2 0.x y
csc , 0r yy
sec , 0r xx
cot , 0x yy
y
x
r
(x, y)
6
Example:
2 55
n45
si 6yr
45s
5o 3 5c x
r
6an 23
t yx
Determine the exact values of the six trigonometric functions of the angle .
6sc
245 5c r
y
4ec 53
5s rx
cot 0.536
xy
2 2 2 2
( )
43
3
6 5
6
9
r x y
y
x
(3, 6)
So, we know that trigonometric function values are side length relationships of right triangles.
We can easily evaluate the exact values of trigonometric functions for special angles.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
8
60○ 60○
Consider an equilateral triangle with each side of length 2.
The perpendicular bisector of the base bisects the opposite angle.
The three sides are equal, so the angles are equal; each is 60.
Geometry of the 30-60-90 triangle
2 2
21 1
30○ 30○
3
Use the Pythagorean Theorem to find the length of the altitude, . 3
Special right triangle relationships
1
6045
3 22
1
1
Now, let’s apply it to the unit circle…
What does “unit circle” really mean?
It’s a circle with a radius of 1 unit.
What is the equation of the “unit circle”?
122 yx
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
, 180 0, 02, 360
3
2
2
Let’s begin with an easy family…4
2
2
2
2
1
45
2
2,
2
2
What are the coordinates?
4
Now, reflect the triangle to the second quadrant…
What are the coordinates?
, 180 0, 02, 360
3
2
2
2
2
2
2
1
45
2
2,
2
2
4
Now, reflect the triangle to the third quadrant…
1
-2
2
2
2
-2
2,
2
2 3
4
What are the coordinates?
Now, reflect the triangle to the fourth quadrant…
, 180 0, 02, 360
3
2
2
2
2
2
2
1
45
2
2,
2
2
4
1
-2
2
2
2
-2
2,
2
2 3
4
-2
2, -
2
2 5
4
What are the coordinates?
, 180 0, 02, 360
3
2
2
2
2
2
2
1
45
2
2,
2
2
4
1
-2
2
2
2
-2
2,
2
2 3
4
-2
2, -
2
2 5
42
2, -
2
2 7
4
30
1 1
2
3
2
6
3
2,1
2
Now, reflect the triangle to the second quadrant.
Complete the family… .6
30
1 1
2
3
2
6
3
2,1
2
Now, reflect the triangle to the third quadrant.
5
6
1
2
-3
2
-3
2,1
2
30
1 1
2
3
2
6
3
2,1
2
Now, reflect the triangle to the fourth quadrant.
5
6
1
2
-3
2
-3
2,1
2
-3
2, -
1
2
What are the coordinates? 7
6
What are the coordinates?
30
1 1
2
3
2
6
3
2,1
2 5
6
1
2
-3
2
-3
2,1
2
-3
2, -
1
2 7
6
3
2, -
1
2
11
6
Let’s look at another “family”3
1
, 180 0, 02, 360
3
2
2
60
3
2
1
2
1
2,
3
2
3
Now, reflect the triangle to the second quadrant
1
, 180 0, 02, 360
3
2
2
60
3
2
1
2
1
2,
3
2
3
Now, reflect the triangle to the third quadrant
1
-1
2
What are the coordinates?
-1
2,
3
2 2
3
3
2
Now, reflect the triangle to the fourth quadrant
1
, 180 0, 02, 360
3
2
2
60
3
2
1
2
1
2,
3
2
3
13
2
-1
2
-1
2,
3
2 2
3
What are the coordinates?
-1
2, -
3
2 4
3
1
, 180 0, 02, 360
3
2
2
60
3
2
1
2
1
2,
3
2
3
13
2
-1
2
-1
2,
3
2 2
3
-1
2, -
3
2 4
3
1
2, -
3
2
What are the coordinates?
5
3
24
x
y (0, 1)
90°2
(–1, 0)
180°
(0, –1)
270°32
360°2 (1, 0)0° 0
Ordered pairs of special angles around the Unit Circle
60° 3
31 ,2 2
45°4
2 2,2 2
30° 6
3 1,2 2
330°116
3 1,2 2
315°74
2 2,2 2
300°53
31 ,2 2
23
120°
31 ,2 2
34
135°
2 2,2 2
56
150°
3 1,2 2
210°76
3 1,2 2
225°5
4
2 2,2 2
240°43
31 ,2 2
sin1y
cos1x
Since r = 1…
cos ,sin ,x y
• Important point:
Since r = 1… sin1y cos
1x
,x y cos ,sin
122 yxBecause ordered pairs around the unit circle (x, y) represent sine and cosine, and the equation of the circle is ,
We have the following identity: 2 2cos sin 1
What if the radius is not 1?
30
30
61
Trigonometric values are functions of the angle – ratios of sides of similar triangles remain the same. So it always holds that .2 2cos sin 1
27
x
y (0, 1)
90°2
(–1, 0)
180°
(0, –1)
270°32
360°2 (1, 0)0° 0
Trigonometric Values of Special Angles
60° 3
31 ,2 2
45°4
2 2,2 2
30° 6
3 1,2 2
330°116
3 1,2 2
315°74
2 2,2 2
300°53
31 ,2 2
23
120°
31 ,2 2
34
135°
2 2,2 2
56
150°
3 1,2 2
210°76
3 1,2 2
225°5
4
2 2,2 2
240°43
31 ,2 2
28
DomainThe domain of the sine and cosine function is the set of all real numbers.
(1, 0)(–1, 0)
(0, –1)
(0, 1)
x
y
x
1 1x
1 1y
Unit Circle
RangeThe point (x, y) is on the unit circle, therefore the range of the sine and cosine function is between – 1 and 1 inclusive.
29
A function f is periodic if there is a positive real number c such that f (t + c) = f (t)
for all t in the domain of f. The least number c for which f is periodic is called the period of f.
x
y
x
Unit Circle
t = 0, 2, …Periodic Function
( ) sinf t t
Period
30
Example:Evaluate sin 5 using its period.
5 - 2 - 2 = sin 5 = sin = 0
Adding 2 to each value of t in the interval [0, 2] completes another revolution around the unit circle.
sin( 2 ) sint n t cos( 2 ) cost n t
( ) sinf t t
x
y
x(–1, 0)
You Try:
a) Evaluate sin
b)Evaluate cos10
3
9
4
Can you?
5tan
6
3sec
4
Evaluate each of the following. Exact values only please.
Even and Odd Trig FunctionsRemember:
if f(-t) = f(t) the function is evenif f(-t) = - f(t) the function is odd
The cosine and secant functions are EVEN.
cos(-t)=cos t sec(-t)=sec t
The sine, cosecant, tangent, and cotangent functions are ODD.
sin(-t)= -sin t csc(-t)= -csc t
tan(-t)= -tan t cot(-t)= -cot t
(1, 0)(–1, 0)
(0,–1)
(0,1)
x
y
x
34
Example:Evaluate the six trigonometric functions at = .
sin 0y
1cos x 0an 01
t yx
1 is un1 defined.csc0y
1 1 11
secx
1 is uc ndefine .o
0t dx
y
(1, 0)(–1, 0)
(0, –1)
(0, 1)
x
y
x
Application:
A ladder 20 feet long leans against the side of a house.The angle of elevation of the ladder is 60 degrees. Find the height from the top of the ladder to the ground.
Application:
An airplane flies at an altitude of 6 miles toward a pointdirectly over an observer. If the angle of elevationfrom the observer to the plane is 45 degrees, find the horizontal distance between the observer and the plane.
.
Homework
4.2 pg. 264 1-51 odd
Trig Races
csc7
22sin
3
cos43
tan 210
csc11
4cos
3
7sin
3
3sin
2
4sec
3
3csc
4
7tan
3
8cot
3
1
2
3
3
2
undef.
3
2
1
2
3
2
1
2
2
3
3
3
HWQ 10/15
5tan
6
csc73sec
4
22sin
3
Evaluate each of the following. Exact values only please.
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