Unit 6 – Introduction to Trigonometry The Unit Circle (Unit 6.3)

Unit 6 – Introduction to Trigonometry
The Unit Circle (Unit 6.3)
William (Bill) Finch
Mathematics Department
Denton High School
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Lesson Goals
When you have completed this lesson you will:
W. Finch
Unit Circle
I
Find values of trigonometric functions for any angle.
I
Find the values of trigonometric functions using the unit
circle.
DHS Math Dept
2 / 25
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Trigonometric Functions of Any Angle
θ is an angle in standard position,
P(x, y ) is a point on the terminal
side, and r is the distance from P to
the origin (denominators 6= 0):
sin θ =
y
r
csc θ =
r
y
cos θ =
x
r
sec θ =
r
x
tan θ =
y
x
cot θ =
x
y
y
P(x, y )
r
θ
x
Recall the equation of a
circle centered at the
origin:
r2 = x2 + y2
p
r = x2 + y2
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DHS Math Dept
Unit Circle
Introduction
3 / 25
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Example 1
The point (−3, 2) is on the terminal side of an angle in
standard position. Find the exact values of the six
trigonometric functions of θ.
W. Finch
Unit Circle
DHS Math Dept
4 / 25
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Quadrental Angles
Quadrental angles terminate on an axis.
y
y
(0, r )
y
y
θ
θ
θ
θ
(r , 0)
x
x
x
(−r , 0)
x
(0, −r )
θ = 0◦ or
0 radians
θ = 90◦ or
π/2 radians
θ = 180◦ or
π radians
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θ = 270◦ or
3π/2 radians
DHS Math Dept
Unit Circle
Introduction
5 / 25
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Example 2
Find the exact value of each trigonometric function, if defined.
If not defined, write undefined.
a) cos π
b) tan(−270◦ )
c) sec
3π
2
d) sin 5π
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Unit Circle
DHS Math Dept
6 / 25
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Angles Not Acute or Quadrental
Quadrant II
y
(−a, b)
b
b
r
a
cos θ = −
r
b
tan θ = −
a
sin θ =
r
θ
θ0
a
x
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b
r
a
cos θ0 =
r
b
tan θ0 =
a
sin θ0 =
DHS Math Dept
Unit Circle
Introduction
7 / 25
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Angles Not Acute or Quadrental
Quadrant III
y
b
r
a
cos θ = −
r
b
tan θ =
a
sin θ = −
θ
a
b
(−a, −b)
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Unit Circle
θ0
r
x
b
r
a
cos θ0 =
r
b
0
tan θ =
a
sin θ0 =
DHS Math Dept
8 / 25
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Angles Not Acute or Quadrental
Quadrant IV
y
b
r
a
cos θ0 =
r
b
tan θ0 =
a
b
sin θ = −
r
a
cos θ =
r
b
tan θ = −
a
θ
a
x
θ0
r b
(a, −b)
sin θ0 =
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DHS Math Dept
Unit Circle
9 / 25
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Reference Angles
If θ is an angle in standard position, its reference angle θ0 is the
acute angle formed by the terminal side of θ and the x-axis.
y
y
θ
x
θ0
y
θ
θ
x
x
θ0
θ0 = θ
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Unit Circle
θ0 = 180◦ − θ
θ0 = π − θ
y
θ0 = θ − 180◦
θ0 = θ − π
θ
x
θ0
θ0 = 360◦ − θ
θ0 = 2π − θ
DHS Math Dept
10 / 25
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Example 3
Sketch the angle and the identify its reference angle.
a) −150◦
b) 315◦
c)
3π
4
d)
5π
3
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DHS Math Dept
Unit Circle
Introduction
11 / 25
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Evaluating Trigonometric Functions of Any Angle
y
1. Sketch the angle.
2. Determine the reference angle θ0 .
3. Find the value of the trig
function for θ0 .
4. Determine the sign (pos or neg)
based on the quadrant
containing the terminal side of θ.
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Unit Circle
Quad
sin θ :
cos θ :
tan θ :
II
+
−
−
Quad I
sin θ : +
cos θ : +
tan θ : +
Quad III
sin θ : −
cos θ : −
tan θ : +
Quad IV
sin θ : −
cos θ : +
tan θ : −
x
DHS Math Dept
12 / 25
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Special Reference Angles
θ (radians)
π
6
π
4
π
3
θ (degrees)
30◦
45◦
√
2
2
√
2
2
60◦
√
3
2
1
2
√
3
sin θ
cos θ
tan θ
1
2
√
3
2
√
3
3
1
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DHS Math Dept
Unit Circle
Introduction
13 / 25
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Example 4
Find the exact value of each expression.
4π
a) sin
3
b) sec
15π
4
c) tan 150◦
d) cos (−120◦ )
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Unit Circle
DHS Math Dept
14 / 25
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Example 5
√
29
Let sec θ =
, where sin θ > 0. Find the exact values of
5
the remaining five trigonometric functions of θ.
W. Finch
DHS Math Dept
Unit Circle
Introduction
15 / 25
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
The Unit Circle
A unit circle is a circle of
radius 1 centered at the origin.
The radian measure of a
central angle is
s
s
θ= = =s
r
1
so the arc length intercepted
by θ equals the angle’s radian
measure.
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Unit Circle
y
(0, 1)
r
s
θ
(−1, 0)
x
(1, 0)
r
(0, −1)
DHS Math Dept
16 / 25
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
The Unit Circle and the Wrapping Function
Place a number line vertically
tangent to a unit circle at (1, 0).
Wrap this line around the circle
(counterclockwise for positive values
and clockwise for negative values),
each point t on the line would map to
a unique point P(x, y ) on the circle.
This is referred to as the wrapping
function w (t). Since r = 1, the six
trigonometric rations of angle t can
be defined in terms of just x and y .
y
P(x, y )
t
t
x
(1, 0)
1
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DHS Math Dept
Unit Circle
Introduction
17 / 25
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Trigonometric Functions on the Unit Circle
sin t = y
cos t = x
y
tan t =
x
And, of course, no
denominator = 0.
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Unit Circle
1
csc t =
y
1
x
x
cot t =
y
sec t =
P(x, y )
P(cos t, sin t)
y
t
t
x
1
These functions are referred to
as circular functions
DHS Math Dept
18 / 25
Introduction
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
16-Point Unit Circle
y
√
−
2
,
2
√
−
3 1
, 2
2
− 12 ,
√ √
3
2
(0, 1)
2
2
√
3
2
√
180◦
π
3
3 1
, 2
2
5π
4 4π
3
5π
3
3π
2
√
225◦
− 23 , − 12
240◦
√
√ 2
2
−2,−2
√ 1
3
−2, − 2
7π
4
π
6
(1, 0)
x
360
0◦ ◦
2π
7π
6
30◦
π
4
π
210◦
2
2
√
45◦
π
2
2π
3π 3
4
√
2
,
2
60◦
135◦
5π
6
1
,
2
90
120◦
(−1, 0)
◦
150◦
11π
6
330◦
√
315◦
300◦
3
,
2
√
◦
270
(0, −1)
1
,
2
√
−
√
2
,
2
3
2
−
2
2
− 21
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DHS Math Dept
Unit Circle
19 / 25
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
16-Point Unit Circle
y
y
√
−
2
,
2
− 12 ,
√ √
3
2
2
2
√
−
◦
π
210
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Unit Circle
225◦
(0, 1)
(0, 1)
90◦
90135◦ ◦
120◦
150◦
5π
6
3π
4
2π
3
2
2
√
−
2
,
2
√
240◦
− 2
2
√ − 12 , − 23
5π
4 4π
3
3π
2
π
2
60◦
√ 1
3
√ 2 , 2
√ √ 1
2
, 23
, 2
2
√ √2 √2 60◦
2
2 3 1
,
45◦2 , 2 2 2
π
3
45◦
π
(−1, 0)2π
◦
π
3
3π 3 180
4
5π
7π
6
6
5π
4 4π
210◦
3
√
225◦
− 3, −1
7π
6
◦
3
2
π
2
1
3 1
, 2
2
√
2
2
135
180
3
2
,
2
◦
150◦
1, 0)
√
−
− 12 ,
√ 120◦
√
− 23 , 12
√
Trig Functions – Circle
π
4
π
4
30◦ √
3 1
, 2
2
π
6
◦
302π
π
6
5π
3
3π
2
2π
7π
4
(1, 0)
x
360
0◦ ◦
11π
6
330◦
315◦◦ ◦
360
0
300◦
(1,
0)
√
3
, −1
2 x2
√ 2
, − 22
2
√ √
270◦
11π
6
7π (0, −1)
4
5π
3
1
,
2
−
3
2
◦
330
315◦
√
DHS Math Dept
3
1
20 / 25
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Unit Circle
Periodic Functions
Summary
Example 6
Find the exact value of each expression.
7π
a) sin
6
b) cos
π
3
c) tan
4π
3
d) sec 270◦
W. Finch
DHS Math Dept
Unit Circle
Introduction
21 / 25
Trig Functions – Circle
Quadrental Angles
Other Angles
Summary
Periodic Functions
A number line can be wrapped around a circle infinitely many
times, so the domain of both the sine and cosine functions is
(−∞, ∞). This means more than one value t will be mapped
onto the same point P(x, y ). Graphing ordered pairs of the
form (t, sin t) shows how the function repeats periodically.
W. Finch
Unit Circle
DHS Math Dept
22 / 25
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Periodic Functions
A function y = f (t) is periodic if there exists a positive real
number c such that f (t + c) = f (t) for all values of t in the
domain of f .
The smallest number c for which f is periodic is called the
period of f .
sin(t + n · 2π) = sin t
period = 2π
cos(t + n · 2π) = cos t
period = 2π
tan(t + n · π) = tan t
period = π
W. Finch
DHS Math Dept
Unit Circle
Introduction
23 / 25
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
Example 7
Use the period of each function to determine an exact value.
9π
a) cos
4
b) sin
c) tan
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Unit Circle
−2π
3
29π
6
DHS Math Dept
24 / 25
Introduction
Trig Functions – Circle
Quadrental Angles
Other Angles
Unit Circle
Periodic Functions
Summary
What You Learned
You can now:
W. Finch
Unit Circle
I
Find values of trigonometric functions for any angle.
I
Find the values of trigonometric functions using the unit
circle.
I
Do problems Chap 4.3 #1, 5, 9-31 odd, , 33-37 odd,
43-57 odd, 61-65 odd, 73, 75
DHS Math Dept
25 / 25