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MATH 107
Section 5.3
Unit Circle Approach
Properties of the Trigonometric Functions
y
x
x
yx
r
r
x
y
r
r
y
cottan
seccos
cscsin
Remember the six trigonometric functions
defined using a point (x, y) on the terminal
side of an angle, .
3© 2011 Pearson Education, Inc. All rights reserved
TRIGONOMETRIC FUNCTIONS AND THE UNIT CIRCLE
A circle with radius 1 centered at the origin of a rectangular coordinate system is a unit circle.
In a unit circle, s = rθ = 1·θ = θ, so the radian measure and the arc length of an arc intercepted by a central angle in a unit circle are numerically identical.
The correspondence between real numbers and endpoints of arcs on the unit circle is used to define the trigonometric functions of real numbers, or the circular functions.
4© 2011 Pearson Education, Inc. All rights reserved
Let t be any real number and let P = (x,y) be the point on the unit circle associated with t. Then
UNIT CIRCLE DEFINITIONSOF THE TRIGONOMETRIC FUNCTIONS
OF REAL NUMBERS
yt sin )0( 1
csc yy
t
xt cos )0( 1
sec xx
t
)0( tan xx
yt )0( cot y
y
xt
We will first look at the special angles called the quadrantal angles.
90
180
270
0
The quadrantal anglesare those angles that lie on the axis of the Cartesian coordinate
system: 0° , 90°, 180°, 270° and 360° .
We can count the quadrantal angles in terms of . radians2
radians2
radians2
2
radians2
3
radians
360,0
90
180
270
radians57.1
radians14.3
radians71.4
radians 2
radians, 0
radians,0
radians28.6
radians2
0 radians
radians2
3
radians
radians2
0
90
180
270
360
(1, 0)
or
undefinedis0cot01
00tan
10sec10cos
undefinedis0csc00sin
radians2
0 radians
radians2
3
radians
radians2
0
90
180
270
360or
(0, 1)
02
cotundefinedis2
tan
undefinedis2
sec02
cos
12
csc12
sin
(-1, 0)
undefinediscot0tan
1sec1cos
undefinediscsc0sin
(0, -1)
02
3cotundefinedis
2
3tan
undefinedis2
3sec0
2
3cos
12
3csc1
2
3sin
Now let’s cut each quadrant in half,
which basically gives us 8 equal
sections.
0
4
4
2
4
4
4
6
4
3
4
5
4
7
4
8We can again count around the
circle, but this time we will count in
terms of
radians.
4
.4
8,
4
7,
4
6,
4
5,
4
4,
4
3,
4
2,
4
1 and
4
2
2
2
3
Then reduce appropriately.
45
90
135
180
225
270
315
360
2
45
45 The lengths of the legs of the 45 – 45 – 90
triangle are equal to each other because their
corresponding angles are equal.
If we let each leg have a length of 1, then we find
the hypotenuse to be using the Pythagorean
theorem.
2
1
1
2
You should memorize this triangle or at
least be able to construct it. These angles
will be used frequently.
Next we will look at two special triangles: the 45 – 45 – 90 triangle and the 30 – 60 – 90 triangle.
These triangles will allow us to easily find the trig functions of the special angles, 45 , 30 , and 60 .
(Knowing this derivation is not necessary,
but knowing the ratio of sides and angles is.)
45
45
1
1
2
145cot145tan
245sec2
245cos
245csc2
2
2
145sin
Using the definition of the trigonometric functions as the ratios of the sides of a right triangle,
we can now list all six trig functions for a angle.45
For the 30 – 60 – 90 triangle, we will construct an equilateral triangle (a triangle with 3 equal angles of
each, which guarantees 3 equal sides).60
If we let each side be a length of 2, then cutting
the triangle in half will give us a right triangle
with a base of 1 and a hypotenuse of 2. This
smaller triangle now has angles of 30, 60, and
90 .
We find the length of the other leg to be using
the Pythagorean theorem.
3
3
60
1
2
30
You should memorize this triangle or at least
be able to construct it. These angles, also,
will be used frequently.(Knowing this derivation is not necessary,
but knowing the ratio of sides and angles is.)
Again, using the definition of the trigonometric functions as the ratios of the sides of a right triangle, we
can now list all the trig functions for a 30 angle and a 60 angle.
330cot3
3
3
130tan
3
32
3
230sec
2
330cos
230csc2
130sin
3
3
3
160cot360tan
260sec2
160cos
3
32
3
260csc
2
360sin
60
30
1
32
All ISine II
III
Tangent
IV
Cosine
I. All
II. Students
III. Take
IV. Calculus
Positive Values for Trigonometric Functions
Note: Because they are reciprocals, the sign of cosecant matches the sign of sine,
the sign of secant matches the sign of cosine, and the sign of cotangent matches
the sign of tangent. Always.
15© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 4 Determining the Quadrant in Which an Angle Lies
Solution
If tan θ > 0 and cos θ < 0, in which quadrant does θ lie?
Because tan θ > 0, θ lies either in quadrant I or in quadrant III. However, cos θ > 0 for θ in quadrant I; so θ must lie in quadrant III.
16© 2011 Pearson Education, Inc. All rights reserved
DEFINITION OF A REFERENCE ANGLE
Let be an angle in standard position that is not a quadrantal angle.
The reference angle for is the positive acute angle (“theta prime”) formed by the terminal side of and the x-axis.
17© 2011 Pearson Education, Inc. All rights reserved
TRIGONOMETRIC FUNCTION VALUES OF COTERMINAL ANGLES
360sinsin n n 2sinsin
360coscos n
These equations hold for any integer n.
in degrees in radians
n 2coscos
18© 2011 Pearson Education, Inc. All rights reserved
DEFINITION OF A REFERENCE ANGLE
Quadrant I Quadrant II
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DEFINITION OF A REFERENCE ANGLE
Quadrant III Quadrant IV
20© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 6 Identifying Reference Angles
Find the reference angle for each angle .
a. = 250º b. = c. =
5.75Solution
a. Because 250º lies in quadrant III, = º. So = 250º 180º = 70º.
• Because lies in quadrant II, = π .
So = π
21© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 6 Identifying Reference Angles
Solution continued
c. Since no degree symbol appears in θ = 5.75,
has radian measure. Now ≈ 4.71 and
2π ≈ 6.28. So lies in quadrant IV and
= 2π . So
= 2π – ≈ 6.28 – 5.75 = 0.53.
22© 2011 Pearson Education, Inc. All rights reserved
USING REFERENCE ANGLES TO FIND TRIGONOMETRIC FUNCTION VALUES
Step 1 Assuming that > 360º or θ < 0°, find a coterminal angle for with degree measure between 0º and 360º. Otherwise, go to Step 2.
Step 2 Find the reference angle for the angle resulting from Step 1. Write the trigonometric function of .
23© 2011 Pearson Education, Inc. All rights reserved
USING REFERENCE ANGLES TO FIND TRIGONOMETRIC FUNCTION VALUES
Step 3 Choose the correct sign for the trigonometric function value of θ based on the quadrant in which it lies. Write the given trigonometric function of θ in terms of the same trigonometric function of θ with the appropriate sign.
24© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 8 Using the Reference Angle to Find Values of Trigonometric Functions
30º360º
3ta
330º
n tan 30º3
Find the exact value of each expression.
Step 1 0º < 330º < 360º; find its reference angle.
Step 2 330º is in Q IV; its reference angle is
a.Solution
.
25© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 8 Using the Reference Angle to Find Values of Trigonometric Functions
3tan t 3330 0ºnº a
3
Solution continued
Step 3 In Q IV, tan θ is negative, so
b. Step 1
11
6
is between 0 and 2π coterminal with
.
.
26© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 8 Using the Reference Angle to Find Values of Trigonometric Functions
1159 2 3sec sec
66 6sec
3
Solution continued
Step 3 In Q IV, sec θ > 0; so
Step 2
2
2 3sec sec
6 3
6 6
11
is in Q IV; its reference angle is
.
.
y
x
330
3
60
1
2
30 AS
T C
3330cot3
3
3
1330tan
3
32
3
2330sec
2
3330cos
2330csc2
1330sin
Example 1 Continued: The six trig functions of 330 are:
30
Your reference angle is 30 .
y
x
Example 1: Find the six trig functions of 330 .
First draw the 330 degree angle.
3
60
1
2
30
AS
T C
330
30
(Answers on next page.)
Your reference angle is 30 .
y
x
Example 2: Find the six trig functions of .
3
60
1
2
30
3
4
First determine the location of .
3
4
3
3
2
3
3
3
3
3
4
3
(Answers on next page.)
Your reference angle is .3
3
60
1
2
30
AS
T C
Example 2: Find the six trig functions of . 3
4
y
x
3
3
2
3
4
3
3
3
3
1
3
4cot3
3
4tan
23
4sec
2
1
3
4cos
3
32
3
2
3
4csc
2
3
3
4sin
Your reference angle is .
3
0 radians
Example 3: Find the exact value of cos .
4
5
We will first draw the angle to determine the quadrant.
4
5
4
4
2
4
3
4
4
AS
T C
45
45
1
12
4
Your reference angle is .4
0 radians
Problem 3: Find the sin .
AS
T C
6
5
6
6
2
6
36
4
6
5
is the reference angle.6
6
0 radians
Problem 7: Find the exact value of cos .
We will first draw the angle to determine the quadrant.
AS
T CNote that the reference angle is .
4
4
13
4
4
2
4
4
4
64
5 4
7
4
8
4
94
10
4
114
3We see that the angle is located in the
3rd quadrant and the cos is negative in the
3rd quadrant.
4
13
4
12
4
13
4
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EXAMPLE 5 Evaluating Trigonometric Functions
Solution
3tan
3
2 2
y
x
Since tan θ > 0 and cos θ < 0, θ lies in Quadrant III; both x and y must be negative.
35© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 5 Evaluating Trigonometric Functions
Solution continued
With , , and we can find
sin
132
and s c .
3
e
rx y
3
1
3 13si
3n
13r
y
13 13se
2c
2x
r
,