1. INTRODUCTION TO TRIGONOMETRIC FUNCTIONS
Unit 33
INTRODUCTION TO TRIGONOMETRIC FUNCTIONS

2. 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

3. EXAMPLES OF IDENTIFYING SIDES
The two right triangles below each have their sides labeled according to a given reference angle
Note: The opposite and adjacent sides vary depending on the reference angle while the hypotenuse stays in the same position in both cases.
Hypotenuse B
Opposite
Adjacent
Hypotenuse A
Opposite
Adjacent

4. TRIGONOMETRIC FUNCTIONS
Three trigonometric functions are defined in the table below
Note: The symbol  denotes the reference angle
FUNCTION
SYMBOL
DEFINITION
sine θ
sin θ
cosine θ
cos θ
tangent θ
tan θ

5. TRIGONOMETRIC FUNCTIONS
Reciprocal trigonometric functions are defined in the table below
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
FUNCTION
SYMBOL
DEFINITION
cotangent 
cot 
secant 
sec 
cosecant 
csc 

6. 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 1 a b c
sin 1 =
csc 1 =
cos 1 =
sec 1 =
tan 1 =
cot 1 =

7. 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, , , , is pressed
sin
cos
tan OR
2. The appropriate function key, , , , is pressed first, and then the value of the angle is entered
sin
cos
tan

8. DETERMINING FUNCTIONS OF ANGLES
Procedure for determining the cosecant, secant, and cotangent functions. (Note: Since these functions are reciprocal functions, the reciprocal key ( , ) must be used on the calculator. The two most common procedures are given below)
1/x
x –1
1. Enter the value of the angle, press the appropriate function key, , , ; press
sin
cos
tan
1/x
2. Press the appropriate function key, , , ; enter the value of the angle; press ; press
sin
cos
tan =
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.

9. 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

10. 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 , , and
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
sin
cos
tan

11. EXAMPLES
Find the angle whose tangent is Round the answer to two decimal places:
Find the angle whose secant is Round the answer to two decimal places:
(or )  or 61.93° Ans
2nd
shift
tan–1
or or 61.93° Ans
shift
tan–1 =
(or )  or 29.79° Ans
1/x
2nd
shift
cos–1
or (or ) ° Ans
shift
cos–1
1/x
x –1 =

12. PRACTICE PROBLEMS
The sides of the triangle below are labeled with different letters. State the ratio of each of the six functions in relation to 1. 1 a b c

13. PRACTICE PROBLEMS (Cont)
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°

14. PRACTICE PROBLEMS (Cont)
Find each angle rounded to the nearest tenth of a degree.
cos A =
tan A =
sec A =
csc A =
cot A =

15. PROBLEM ANSWER KEY
2. (a) (b) (c) (d) (e)
3. (a) 47.6° (b) 51.3° (c) 75.6° (d) 21.2° (e) 75.2°