Appendix D: Trigonometry Review

A triangle in which one angle is a right angle is called a

right triangle

. The side opposite the right angle is called the

hypotenuse

, and the remaining two sides are called the

legs

of the triangle.

c b θ a

Acute Angle 

Initial Side c



a b

Right Triangle

Six Trigonometric Ratios Formal Name

Cosine of θ Sine of θ Tangent of θ

Abbreviation

cos θ sin θ tan θ

b c a θ

Ratio

adjacent hypotenuse



a c opposite hypotenuse



b c opposite adjacent



b a

Six Trigonometric Ratios Formal Name

Secant of θ Cosecant of θ Cotangent of θ

Abbreviation

sec θ csc θ cot θ

b c a θ

Ratio

hypotenuse adjacent



c a hypotenuse opposite



c b adjacent opposite



a b

12

Example: Find the value of each of the six trigonometric functions of the angle θ.

 13

c

= Hypotenuse = 13

a

= Adjacent = 12

a

2 +

b

2 =

c

2 12 2 +

b

2 = 13 2

b

2 = 13 2 - 12 2

b

= 5

a

= Adjacent = 12

b

= Opposite = 5

c

= Hypotenuse = 13 cos  

adj hyp

 12 13 sin  

opp hyp

 5 13 tan  

opp adj

 5 12 12  sec  

hyp adj

 13 12 csc  

hyp opp

 13 5 tan  

adj opp

 12 5 13

Reciprocal Identities

sec   1 cos  csc   sin 1  cot   1 tan 

Quotient Identities

tan   sin  cos  cot   cos  sin 

The inverse (or arc) trigonometric functions will return an angle for a given real number.

cos  sin  tan     cos  1

b

or   arccos

b

sin  1

b

or   arcsin

b

tan  1

b

or   arctan

b

Use your calculator to evaluate the following. Explain the relationship between the two sets of numbers.

 1 ( ) tan( 15 ); tan ( 0.26795)  1 (

d

) Why doesn’t this relationship hold for  1

Trigonometric Functions

We know that angles (either in degrees or radians) are essentially real numbers (any number you can think of). Therefore, the trigonometric ratios can be thought of as functions where the input is the angle (any real number).

f(x) = cos x Notes:

If a degree symbol appears on the angle measurement, it is given in degrees. If nothing appears after the angle measurement, it is given in radians. Be sure to set your calculator to the appropriate mode.

Coordinates of Points

r x a • P

(

a

,

b

)

b

Determine

a

and

b

as functions of

x

.

cos

x a



a r



r

cos

x

sin

x



b r b



r

sin

x r



a

2 

b

2

Trigonometric Functions

• The triangle formed by dropping a perpendicular line from the terminal side of an angle to the closest side of the

x

-axis is called a

reference triangle

.

• We will not use

x

and

y

to reference the point

P

. The variable

x

is reserved angle measurements.

• It is very important to include the sign with the values of

a

and

b

when labeling your reference triangle. This will assure that the values of the six trigonometric ratios are correct.

Pythagorean Identities

a

2 

b

2 

c

2

a

2

c

2 

b

2

c

2 

c c

2 2

a c

2 

b c

2  1  cos    sin   2  1

c b

 cos   2

a

 cos 2 

θ

cos 2   sin 2   1 sec 2   tan 2   1 csc 2   cot 2   1

Other Identities

• Many more trigonometric identities can be found on page A29 of your text. These identities may be used in this and further calculus classes.

Graphs of Trigonometric Functions

f(x) = cos x

Domain: (-∞, ∞) Range: [-1, 1]

f(x) = sin x

Domain: (-∞, ∞) Range: [-1, 1]

Graphs of Trigonometric Functions

f(x) = tan x

Domain: (-∞, ∞) Range: [-1, 1]