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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
DEFINITIONS:
An angle is defined as the set of points determined by two rays, or half-lines, l1 and l2 having the
same end point O. An angle can also be considered as two finite line segments with a common
point.
We call l1 the initial side, l2 the terminal side, and O the vertex of angle AOB. The direction
and number of rotations of l1 makes before stopping at l2 is not restricted. If two angles have the
same initial and terminal sides, they are coterminal angles.
terminal side
l2
vertex
l2
l1
coterminal angles
l1
initial side
A straight angle is an angle whose sides lie on the same straight line but extend in opposite
directions from its vertex.
If we introduce the rectangular coordinate system, then the standard position of an angle is
obtained by placing the vertex at the origin and letting the initial side l1 coincide with the positive
x-axis. If l1 is rotated in a counterclockwise direction to the terminal position l2, then the angle is
considered positive. If l1 is rotated in a clockwise direction, the angle is negative.
A positive angle in
standard position
A negative angle in
standard position
An angle is called a quadrantal angle if its terminal side lies on a coordinate axis.
One unit of measurement for angle is the degree. The angle is standard position obtained by one
complete revolution in the counterclockwise direction has measure 360 degrees, written 360.
Two coterminal angles will differ by multiples of 360°. To find a coterminal angle, add or
subtract multiples of 360°.
Ex 1: Find two positive coterminal angles and two negative coterminal angles for the angles
below .
 = 55
 = –100
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
A right angle is half of a straight angle and has measure 90.
An acute angle ; 0 <  < 90
An obtuse angle ; 90 <  < 180
Complementary angles , ;  +  = 90 Supplementary angles , ;  +  = 180
Complementary angles have a sum of 90°. Supplementary angles have a sum of 180°.
For smaller units than degrees we have two choices:
1) tenths, hundredths, thousandths, and ten- thousandths of a degree. example: 150.1234
2) divide the degrees into 60 equal parts, called minutes (denoted by ˊ) and each minute into
60 equal parts, called seconds (denoted by ") example: 50 = 49 59' 60"
Ex 2:
Find the angle that is complementary
to  = 45.7
Find the angle that is complementary
to  = 39 34' 19"
Find the angle that is supplementary
to  = 155.41
.
Find the angle that is supplementary
to  = 52 13' 45"
To convert from a decimal part of a degree to minutes and seconds: Multiply the decimal
part of the degree by 60 minutes/degree. The result is the number of minutes. Take the
decimal part of a minute and multiply by 60 seconds/minute and round to nearest number
60
60
of seconds. Multiply by the ratios
. This is DD → DMS!
or
1
1
Ex 3:
Express the angle in terms of degrees, minutes and seconds.
 = 150.1459
 = 12.6789
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
To convert from minutes and seconds to decimal part of a degree: Write x minutes as x/60
of a degree and y seconds as y/3600 of a degree, convert to decimals, add and round.
1
1
Multiply by the ratios
This is DMS→DD
or
60
3600
Express the angle as a decimal to the nearest ten-thousandth.
Ex 4:  = 45 16' 45"
 = 115 50' 12"
Another unit of measurement for angles is the radian. One radian is the measure of the central
angle of a circle subtended by an arc equal in length to the radius of the circle. Since the
circumference of a circle is 2r, the number of times r units can be ‘laid off’ around the circle is
2.
******Thus 360 = 2 radians and therefore 180 =  radians. ******
When the radian measure of an angle is used, no units will be indicated.
 = 5 means  = 5 radians. θ = 5 means 5 degrees.
 radians
180
(since both = 1) to covert degree
180
 radians
measures to radians and radian measures to degrees. To remember which ones to use, think of
which unit needs to be canceled and which unit is to be left as the label.
Since 180 =  radians, use the ratios
Multiply by

180
to convert degrees to radians.
and
Multiply by
180

to convert radians to degrees.
Ex 5: Find the exact radian measure.
ex.
30
ex.
60
ex.
90
ex.
120
ex.
-45
ex.
250
ex.
450
ex.
360
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
Ex 6: Find the exact degree measure.
3
4
5
3
11
6
3
2
Ex 7: Approximate the number of degrees to 3 decimal places.
3 radians
Ex 8:
Express the angle in terms of degrees, minutes and seconds.
=3
 = 5.6
Remember: Coterminal angles vary by
multiples of 360°. Since 360° equals
2π, to find a coterminal angle that is in
radian measure, add or subtract
multiples of 2π.

Two coterminal angles will differ by multiples of 2π radians. To find a coterminal angle,
add or subtract multiples of 2π.
Ex 9:
=
Find two positive coterminal angles and two negative coterminal angles.

3
=
7
6
= 
5
4
4
MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
Remember, when converting from:

degrees to radians: multiply by
180
Ex 10:
 = 2.3
radians to degrees: multiply by
180

Express  in terms of degrees, minutes and seconds.
Find the exact radian measure of .
 = 240
Find the length of arcs and the area of a sector:
If  is in radians then the length of the arc created by  is found by: s = r (From C = 2r)
C  2 r
For a circle: C    2

Therefore: C   r
which means in general s   r
Ex 11:
Find the length of the arc subtended by the central angle.
s
s
60
10 cm
240
11 cm
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
1 2
r  (From A = r2)
2
A   r2
Since  =2 for a circle
If  is in radians then the area of the sector created by  is found by: A =

2

 
Substitute: A    r 2
2
1
or A  r 2
2
Ex 12: Find the area of the sectors subtended by the central angle in the examples at the
bottom of page 5.

Ex 13: Find the radian and degree measures of the central angle  subtended by the given arc
of length s on a circle of radius r.
a) Find the radian and degree measures (DD and DMS) of the central angle .
b) Find the area of the sector determined by 
s = 8 cm, r = 3 cm
s = 6 ft, r = 18 in.
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
Ex 14:
a) Find the length of the arc that subtends the given central angle  on a circle of diameter d.
b) Find the area of sector determined by 
 = 50, d = 16 m
 = 3.1, d = 44 cm
Ex 15:
The minute hand of a clock is 6 inches long. How far does the tip of the minute hand move in 15
minutes? How far does it move in 25 minutes?
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
We will introduce the trigonometric functions in the manner in which they originated
historically- as ratios of sides of a right triangle.
A triangle is a right triangle if one of its angles is a right angle.
Opposite
Hypotenuse

Adjacent
If θ would be changed to the other acute
angle of the triangle, the opposite and
adjacent sides would switch. Labeling
triangles is always important. Changing θ
changes the adjacent and opposite sides.
The hypotenuse is always the same side no
matter where θ is put.
With  as the acute angle of interest, the adjacent side is abbreviated adj., the opposite side is
abbreviated opp., and the hypotenuse is abbreviated hyp. With this notation, the six
trigonometric functions become:
opp
adj
opp
The sine, cosine, and tangent are the 3 major
sin 
cos 
tan 
hyp
hyp
adj
or primary trigonometric functions. The
cosecant, secant, and cotangent functions are
hyp
hyp
adj
csc  
sec  
cot  
the lesser or minor functions.
opp
adj
opp
SOH-CAH-TOA
sine is opposite over hypotenuse
cosine is adjacent over hypotenuse
tangent is opposite over adjacent
Notice that the csc  is the reciprocal of sin , sec  is the reciprocal of cos , and cot  is the
reciprocal of tan .
1
1
1
sin 
cos 
tan 
csc 
sec 
cot 
These are called the RECIPROCAL IDENTITIES.
1
1
1
csc  
sec  
cot  
sin 
cos 
tan
Since the hypotenuse is always the largest side of the triangle,
0  sin   1
0  cos <1
Because the hypotenuse is always the
(in a right triangle)
largest side, any of the functions with
csc   1
sec   1
‘hyp’ in the denominator, will be less
than 1. Any with’ hyp ‘in the
numerator will be greater than 1.
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
Ex 16:
Find the exact values of the six trigonometric functions for the angle 

c
12
7
5

5
a
Ex 17:
Find the exact values of the six trigonometric functions for the acute angle 
cos 
9
41
sec  
5
2
cot  
35
12
csc  5
7
3
tan 

sin 
195
28
9

MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
Ex 18:
Using your calculator, find (approximate to 4 decimal places):
DEGREES
sin(134)
cos(-54)
tan(121)
sec(-94)
csc(25)
cot(330)
sin(5)
cos(-0.123)
6 
tan 
5 
sec(3.1)
csc(5.6)
cot(-9)
RADIANS
10

MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
This is the table of trigonometric values you should be able to recall or derive.
ANGLES
Degrees
Radians
θ
30°
sin 
Reverse the
sine row
cos 
Divide sine row
by cosine row
tan 
θ
45°

6
1
2

4
1  2


2  2 
1  2


2  2 
3
2
1  3


3  3 
Ex 19:
Find the exact values of x and y.
A
θ
60°

3
3
2
45°
30°
1
2
1
2
1
45°
1
60°
3
1
B
C
8
y
y
y
5
x
60
30
x
This table can easily be
verified by examining a 45-45
right triangle and a 30-60
right triangle. (See below.)
x
45
6
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
Always draw a picture with applied problems!
Ex 20:
A building is known to be 500 feet tall. If the angle from where you are standing to the top of the
building is 30°, how far away from the base of the building are you standing?
Ex 21:
The angle to the top of a flag pole is 41°. If you are standing 100 feet from the base of the flagpole, how
tall is the flagpole?
Ex 22:
The angle to the top of a cliff is 57.21°. If you are standing 300 meters from a point directly below the
cliff, how high is the cliff?
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
The Fundamental Identities:
(1) The reciprocal identities:
1
1
csc  
sec  
sin 
cos 
cot  
1
tan
Note:
(sin  ) 2 is
usually
(2) The tangent and cotangent identities:
sin
cos 
tan 
cot  
cos
sin 
(3) The Pythagorean identities:
2
2
2
2
sin   cos  1
1 tan   sec 
written
sin 2  .
1 cot   csc 
2
2
Similar notation
for the other
functions.
The reciprocal identities are obvious from the definitions of the six trigonometric functions.
Take the simple right triangle with sides 3, 4, and 5 with θ opposite the side of length 3.
3
5
csc  
5
3
4
5
cos  
sec  
5
4
3
4
tan  
cot  
4
3
sin  
5
3
θ
4
To prove the tangent identity, examine the following. The cotangent identity proof is similar.
opp
opp opp hyp hyp sin 
tan  




adj hyp adj adj cos 
hyp
3
4
3
cos  
tan  
5
5
4
3
sin  5 3 5 3
   
cos  4 5 4 4
5
sin 
 tan  
cos 
sin  
Use the simple right triangle with sides 3, 4, and 5 on page 13 to find
sin  , cos  , and sin 2   cos2  . This information can be used to prove the Pythagorean
Identities. See the proofs on the next page. sin 2   cos2  is called a Pythagorean identity since
it is derived from the Pythagorean Theorem.
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
3
4
cos  
5
5
2
2
(sin  )  (cos  )  ?
sin  
sin 2   cos 2   ?
2
2
3  4
     ?
5  5
9 16

?
25 25
25
1
25
 sin 2   cos 2   1
x2  y 2  r 2
opp 2  adj 2  hyp 2
x2 y 2 r 2


r2 r2 r2
opp 2 adj 2 hyp 2


hyp 2 hyp 2 hyp 2
cos 2   sin 2   1
sin 2   cos 2   1
Divide both sides of the Pythagorean identity above by sin 2  to get a second Pythagorean
identity.
sin 2   cos 2   1 Divide each side by sin 2 
sin 2  cos 2 
1


2
2
sin  sin  sin 2 
 cos  
2
1 
  csc 
 sin  
1  cot 2   csc 2 
2
To get the remaining Pythagorean identity, divide each side of the identity sin 2   cos2   1 by
cos2  .
Each of the three Pythagorean identities creates two more identities by subtracting a term from
the left side to the right side.
sin 2   cos 2   1
1  tan 2   sec 2 
1  cot 2   csc2
sin 2   1  cos 2 
tan 2   sec 2   1
cot 2   csc2   1
cos 2   1  sin 2 
1  sec 2   tan 2 
1  csc 2   cot 2 
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
Ex 23: Verify each identity by transforming the left side to the right side.
(a)
tan  cot   1
(b) sin(3 )cot(3 )  cos(3 )
(c )
sec 
 csc 
tan 
(e) (1  cos  )(1  cos  )  sin 2 
 
 
sin    cos  
2
 2   1  cot   
(d )
 
 
2
sin  
2
( f ) cos 2  (sec2  1)  sin 2 
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
( g ) (1  sin 2  )(1  tan 2  )  1
(h) cot   tan   csc sec
Below is a picture of an angle θ drawn in standard position (vertex at the origin and the initial
side on the positive x-axis.
(x, y)
y= opposite

x = adjacent
Ex 24:
If θ is an angle is standard position on a rectangular coordinate system and if P(5,12) is on the
terminal side of θ, find the values of the six trigonometric function of θ. (Hint: Draw a picture.)
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
Ex 25:
If θ is an angle in standard position on a rectangular coordinate system and if P(4,3) is on the
terminal side of θ, find the values of the six trigonometric functions of θ.
Ex 26:
Find the exact values of the six trigonometric functions of θ, if θ is in standard position and the
terminal side of θ is in quadrant III and on the line with equation 4 x  3 y  0 .
Notice:
tan   slope of the line
Ex 27:
Find the exact values of the six trigonometric functions of θ, if θ is in standard position and the
terminal side of θ is in quadrant II and parallel to the line with equation 3x  y  7  0
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MA 15800
Lesson 19
Summer 2016
Angles and Trigonometric Functions
Ex 28: Find the quadrant containing θ if the given conditions are true.
a) tan  < 0 and cos  > 0
b)
sec  > 0 and tan  < 0
c)
csc  > 0 and cot  < 0
d)
cos  < 0 and csc  < 0
e)
cos θ < 0 and sec θ > 0
I
II
sin +
csc
tan
III cot +
All +
cos
sec +
IV
To help you remember the picture above: Think (in order of
quadrants): ALL STUDENTS TAKE CALCULUS.
Quadrant I: All functions are positive values.
Quadrant II: Sine and its inverse are positive, others negative.
Quadrant III: Tangent and its inverse are positive, others negative.
Quadrant IV: Cosine and its inverse are positive, other negative.
Ex 29:
Use the fundamental identities to find the values of the trigonometric functions for the given
conditions below
12
and cos   0
5
(a)
tan  
(b)
sec  4 and csc  0
18