Unit 33
INTRODUCTION TO
TRIGONOMETRIC
FUNCTIONS
IDENTIFYING THE SIDES OF A
RIGHT TRIANGLE





The sides of a right triangle are named the opposite
side, adjacent side, and hypotenuse
The hypotenuse is the longest side of a right triangle
and is always opposite the right angle
The positions of the opposite and adjacent sides
depend on the reference angle
The opposite side is opposite the reference angle
The adjacent side is next to the reference angle
2
EXAMPLES OF IDENTIFYING SIDES

The two right triangles below each have their
sides labeled according to a given reference
angle
Adjacent

A
Adjacent
B
Opposite
Note: The opposite and adjacent sides vary depending on the
reference angle while the hypotenuse stays in the same
position in both cases.
3
TRIGONOMETRIC FUNCTIONS

Three trigonometric functions are defined in the table
below
FUNCTION SYMBOL
sine θ
sin θ

DEFINITION
opposite side
hypotenuse
cosine θ
cos θ
adjacent side
hypotenuse
tangent θ
tan θ
opposite side
adjacent side
Note: The symbol  denotes the reference angle
4
TRIGONOMETRIC FUNCTIONS

Reciprocal trigonometric functions are defined in the
table below
FUNCTION


SYMBOL
DEFINITION
cotangent 
cot 
adjacent side
opposite side
secant 
sec 
hypotenuse
adjacent side
cosecant 
csc 
hypotenuse
opposite side
Note: The symbol  denotes the reference angle
These are not used as much due to the fact they are not easily
available on your calculator
5
RATIO EXAMPLE

The sides of the triangle below are labeled
with different letters, then each of the six
trigonometric functions are given using 1
as the reference angle
c
a
a
sin 1 =
c
c
csc 1 =
a
b
c
a
tan 1 =
b
sec 1 =
cos 1 =
1
b
c
b
b
cot 1 =
a
6
DETERMINING FUNCTIONS


Determining functions of given angles is readily
accomplished using a calculator
Procedure for determining the sine, cosine, and
tangent functions: (Note: The procedure for determining
functions of angles varies with different calculators; basically,
however, there are two different procedures)
1. The value of the angle is entered first, and then the
appropriate function key, sin , cos , tan , is pressed
OR
2. The appropriate function key, sin , cos , tan , is
pressed first, and then the value of the angle is entered
7
DETERMINING FUNCTIONS OF ANGLES

Procedure for determining the cosecant, secant,
and cotangent functions. (Note: Since these
functions are reciprocal functions, the reciprocal
key ( 1/x , x –1 ) must be used on the
calculator. The two most common procedures
are given below)
1. Enter the value of the angle, press the appropriate
function key, sin , cos , tan ; press 1/x
2. Press the appropriate function key, sin , cos , tan ;
enter the value of the angle; press = ; press 1/x
**As mentioned before, since these are not readily available on your calculator we will
not use these much so do not waste a lot of time figuring this out on your calculator.
8
ANGLES OF GIVEN FUNCTIONS



Determining the angle of a given function is the inverse
of determining the function of a given angle. When a
certain function value is known, the angle can be found
easily
The term arc is often used as a prefix to any of the
names of the trigonometric functions, such as arcsine,
and arctangent. Such expressions are called inverse
functions and they mean angles
Arcsin is often written as sin–1, arccos is written as cos–1,
and arctan is written as tan–1
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DETERMINING ANGLES



The procedure for determining angles of given functions
varies somewhat with the make and model of calculator
With most calculators, the inverse functions are shown
as second functions [sin–1], [cos–1], and [tan–1] of function
keys sin , cos , and tan
With some calculators, the function value is entered
before the function key is pressed. With other
calculators, the function key is pressed before the
function value is entered
10
EXAMPLES

Find the angle whose tangent is 1.875.
Round the answer to two decimal
places:
• 1.875 2nd (or shift ) tan–1  61.927513 or 61.93° Ans
or shift tan–1 1.875 = 61.927513 or 61.93° Ans

Find the angle whose secant is 1.1523.
Round the answer to two decimal
places:
• 1.1523 1/x 2nd (or shift ) cos–1  29.79261 or 29.79° Ans
or shift cos–1 1.1523 1/x (or x –1 ) =
29.79° Ans
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PRACTICE PROBLEMS
1.
The sides of the triangle below are
labeled with different letters. State the
ratio of each of the six functions in
relation to 1.
c
a
1
b
12
PRACTICE PROBLEMS (Cont)
2.
Find the value of each trigonometric ratio
rounded to four significant digits.
a. sin 68.4°
d. sec 7°39'
b. tan 79°15'
e. cot 54.5°
c. csc 80.3°
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PRACTICE PROBLEMS (Cont)
3.
Find each angle rounded to the
nearest tenth of a degree.
a.
b.
c.
d.
e.
cos A =
tan A =
sec A =
csc A =
cot A =
0.6743
1.2465
4.0347
2.7659
0.2646
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PROBLEM ANSWER KEY
a
c
b
cos 1 
c
a
tan 1 
b
1. sin 1 
c
a
c
sec 1 
b
b
cot 1 
a
csc 1 
2. (a) 0.9298
(b) 5.2672
(c) 1.0145
(d) 1.0090
(e) 0.7133
3. (a) 47.6°
(b) 51.3°
(c) 75.6°
(d) 21.2°
(e) 75.2°
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