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Math 1330 Section 4.3
Section 4.3
Trigonometric Functions of Angles
Unit Circle Trigonometry
An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis.
Positive angles are measured counterclockwise from the initial side. Negative angles are measured clockwise.
We will typically use the  to denote an angle.
Example 1: Draw each angle in standard position.
a. 240
c.
7
3
b.  150
d.
 5
4
Angles that have the same terminal side are called co-terminal angles. Measures of co-terminal angles differ by
a multiple of 360 if measured in degrees or by multiple of 2 if measured in radians.
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Math 1330 Section 4.3
Angles in standard position with the same terminal side are called coterminal angles. The measures of two
coterminal angles differ by a factor corresponding to an integer number of complete revolutions.
The degree measure of coterminal angles differ by an integer multiple of 360 . For any angle  measured in
degrees, an angle coterminal with  can be found by the formula   n *360 .
Example 2: Find three angles, two positive and one negative that are co-terminal with each angle.
A.
40°
B.   

6
Popper 9 question 1: Find the missing side of the following right triangle BAC, if AB =3 and BC =5
A
B
C
A. 9
B. 4
C. 16
D. 25
Popper 9 questions 2: Find cos C.
A.
B.
C.
D
Popper 9 questions 3: Find sin B.
A.
B.
C.
D
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Math 1330 Section 4.3
An angle is formed by two rays that have a common endpoint (vertex). One ray is called the initial side and the
other the terminal side. A terminal angle can lie in any quadrant, on the x-axis or y-axis.
An angle is in standard position if the vertex is at the origin of the two-dimensional plane and its initial side
lies along the positive x-axis.
Positive angles are generated by counterclockwise rotation. Negative angles are generated by clockwise
rotation.
The Reference Angle or Reference Number
Let  be an angle in standard position. The reference angle associated with  is the acute angle (with positive
measure) formed by the x-axis and the terminal side of the angle  . When radian measure is used, the reference
angle is sometimes referred to as the reference number (because a radian angle measure is a real number).
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Math 1330 Section 4.3
Example 3: Draw each angle in standard position and specify the reference angle.
a. 405 o
Note: Notice that in part a. 405 0 terminates in the same position as 45 0 . These type of angles are called
coterminal angles. Every angle has infinitely many coterminal angles. An angle of
x o is coterminal with angles x o + k 360 o = x + 2  k, where k is an integer.
b.  = 330 o
c.  = 
d.  =
2
3
5
4
We previously defined the six trigonometry functions of an angle as ratios of the length of the sides of a right
triangle. Now we will look at them using a circle centered at the origin in the coordinate plane. This circle will
have the equation x 2  y 2  r 2 . If we select a point P(x, y) on the circle and draw a ray from the origin though
the point, we have created an angle in standard position.
4
Math 1330 Section 4.3
Trigonometric Functions of Angles
If the circle is the unit circle, then r = 1 and we get the following:
cos   x
sin   y
tan  
y
x
1
x
1
csc  
y
x
cot  
y
sec  
( x  0)
( x  0)
( y  0)
( y  0)
Example 4: For each quadrantal angle, give the coordinates of the point where the terminal side of the angle
interests the unit circle.
a. - 180 o , then find sine and cotangent, if possible, of the angle.
b.
3
, then find tangent and cosecant, if possible, of the angle.
2
5
Math 1330 Section 4.3
Given a right triangle ABC with B = 90° and tan(A)=
.
Popper 9 question 4: Find sin (A).
A.
√
B.
C. 5 √2
D.
√
Popper 9 question 5: Find cos (C).
A.
√
B.
C. 5 √2
D.
√

or 90
2
cos , sin 
 or 180
0 or 360
3
or 270
2
Recall: (x, y) = (cos  , sin  )
y
(-, +)
All Students Take Calculus
y
(+, +)
S
x
(-, -)
(+, -)
A
OR
x
T
C
This should help you to know which trigonometric functions are positive in which quadrant.
Example 5: Name the quadrant in which the given conditions are satisfied.
a. sin( )  0, cos( )  0
b. tan( )  0, sin( )  0
6
Math 1330 Section 4.3
c. cos( )  0, csc( )  0
d. sec( )  0, cot( )  0
Example 6: Rewrite each expression in terms of its reference angle, deciding on the appropriate sign (positive
or negative).
 7 
a. tan 

 4 
b. sin(140 o )
 2 
c. sec 

 3 
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Math 1330 Section 4.3
Find the trig functions for the angle


or 30 and
or 60 .
3
6
.
sin
 1

6
2
csc

2
6
cos

3

6
2
sec

2
2 3


6
3
3
tan

1
3


6
3
3
cot


6
sin
3

3

3
2
cos
 1

3 2
tan


3
3
csc

2
2 3


3
3
3
sec

2
3
cot

1
3


3
3
3
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Math 1330 Section 4.3
Find the trig functions for the angle
sin
cos

or 45 .
4

2

4
2
csc

2

4
2

2
2 2



4
2
2
sec
2

tan  2  1
4
2
2
Use for questions 1 and 2: If AC =

2
2 2



4
2
2
cot
√
2
2

1
4
:
Popper 10 question 1: Find y:
A. 1
B.
√
C.
D.
√
D.
√
Popper 10 question 2: Find x:
A 1
B.
√
C.
9
Math 1330 Section 4.3
You will need to find the trigonometric functions of quadrant angles and of angles measuring 30,45or 60
without using a calculator.
Here is a simple way to get the first quadrant of trigonometric functions. Under each angle measure, write
down the numbers 0 to 4. Next take the square root of the values and simplify if possible. Divide each value by
2. This gives you the sine values of each of the angles you need. To fine the cosine values, write the previous
line in reverse order. Now find the tangent values by using the sine and cosine values.
Angle in
degrees
Angle in
Radians
0
30

6
0
45

4
60

3
90

2
Sine
Cosine
Tangent
Recall: csc  
1
sin 
sec  
Popper 10 question 3: Suppose that
A.
√
B.
√
C.
1
cos 
cot  
1
tan 
is an acute angle of a right triangle and that
√
D.
√
B. 240°
C. 210°
.
√
Popper 10 Question 4: Convert the following radian measure to degrees:
A. 220°
. Find
.
D. 200°
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Math 1330 Section 4.3
Unit circle
(0, 1)
(-1, 0)
(1, 0)
(0, -1)
Example 7: Let the point P(x, y) denote the point where the terminal side of angle  (in standard position)
meets the unit circle. P is in Quadrant III and y = -5/13. Evaluate the six trig functions of  .
Example 8: Suppose that csc   
9
and that 270 o    360 o . Find cot  and sin  .
4
11
Math 1330 Section 4.3
Evaluating Trigonometric Functions Using Reference Angles
1. Determine the reference angle associated with the given angle.
2. Evaluate the given trigonometric function of the reference angle.
3. Affix the appropriate sign determined by the quadrant of the terminal side of the angle in standard position.
Example 9: Evaluate the following.
a. sin(300 o )
b. cos(-240 o )
c. sec(135 o )
 3 
d. cos  
 4 
 4 


e. sin  3 
12
Math 1330 Section 4.3
f.
 
tan   
 2
g. csc(2  )
h. cot(-5  )
 11 
i. sin  

 2 
Popper 10 question 5: Rewrite the following expression in terms of its reference angle,
deciding on the appropriate sign(positive or negative): cos 120° .
A. cos
120°
B. cos 60°
C.
cos 30°
D.
cos 60°
13
Math 1330 Section 4.3
Solutions to Online Math 1330 Popper 07 1. The graph of the function: . Find the x –coordinate at where the graph crosses the horizontal asymptote (if the graph does cross). A. 2 B. 2 D. Does not cross C. is 3. 2. Find the linear function f with f(2) = 3 and x‐intercept on 3 B. A. 3
3 C. 3
3 D. 3
3 3. State the equations of the asymptotes for the following: A. 4and 1 4 B. D. C. 6and 6 4. Find the value for p: 12 A. 0 B. 3 C. 12 D. 3 5. Find the y‐intercept of the following: 3
1 4 2 A. 0, 0 B. 0, 1 C. 0, 48 D. 0, 48 Solutions for Popper 8 on next page:
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M 1330 Number 08
Grading ID
1 Given the following triangle, solve for x.
x
4
45°
A. 2 2
2.
Given the following triangle, solve for x.
60°
4
x
A. 4 3
2π
π
to degrees.
3
3. Convert
A. 120°
4. Convert 225 ° to radians.
A.
2.
5π
π
4
5. Given the following triangle find the tan B.
A
9
6
C
A. tan B =
B
2 5
5