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
b

a
Right Triangle
Six Trigonometric Ratios
c
b
a
Formal Name
Cosine of θ
Sine of θ
Tangent of θ
θ
Abbreviation
Ratio
cos θ
adjacent
a

hypotenuse c
sin θ
opposite
b

hypotenuse c
tan θ
opposite b

adjacent a
Six Trigonometric Ratios
c
b
a
Formal Name
Secant of θ
Cosecant of θ
Cotangent of θ
θ
Abbreviation
Ratio
sec θ
hypotenuse c

adjacent
a
csc θ
hypotenuse c

opposite
b
cot θ
adjacent a

opposite b
Example: Find the value of each of the six
trigonometric functions of the angle θ.
c = Hypotenuse = 13
a = Adjacent = 12

12
13
a2 + b2 = c2
122 + b2 = 132
b2 = 132 - 122
b=5
a = Adjacent = 12
b = Opposite = 5
c = Hypotenuse = 13

12
adj 12
cos  

hyp 13
hyp 13
sec  

adj 12
opp 5
sin  

hyp 13
hyp 13
csc  

opp 5
adj 12
tan  

opp 5
opp 5
tan  

adj 12
13
Reciprocal Identities
1
sec  
cos 
1
csc  
sin 
1
cot  
tan 
Quotient Identities
sin 
tan  
cos 
cos 
cot  
sin 
The inverse (or arc) trigonometric
functions will return an angle for a
given real number.
cos   b    cos b or   arccos b
1
sin   b    sin b or   arcsin b
1
tan   b    tan b or   arctan b
1
Use your calculator to evaluate the
following. Explain the relationship
between the two sets of numbers.
(a ) cos 45; arccos(0.7071)
1
(b) tan( 15); tan ( 0.26795)
(c) sin(79); sin 1 (0.981627)
(d) Why doesn’t this relationship hold for
tan(215) and tan 1 (.70002)
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
b
x
a
Determine a and b as
functions of x.
P(a, b)
a
cos x 
r
a  r cos x
b
sin x 
r
b  r sin x
r  a b
2
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
c
b
a2  b2  c2
2
2
θ
a
2
a b
c
 2  2
2
c
c
c
2
cos  
2
 cos 
2
2
a b
    1
c c
cos    sin  
2
2
cos 2   sin 2   1
sec   tan   1
2
1
2
csc   cot   1
2
2
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]