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A Review of
Trigonometric
Functions
Right Triangle Vocabulary
opposite
adjacent
B
a
C
c
opposite
adjacent
b
A
Trigonometric Functions

Defined in terms of right triangles
 sin(x) = opp/hyp
 cos(x) = adj/hyp
 tan(x) = opp/adj
= sin(x) / cos(x)

Know the graphs
Trigonometric Functions

Defined in terms of the unit circle
P(x)=(cos x, sin x)
x
cos x
sin x
1
1
Other Trig Functions



cot(x) = 1/tan(x) = cos(x) / sin(x)
sec(x) = 1/cos(x)
csc(x) = 1/sin(x)
Odd/Even

Odd
 Sin(x)
 Csc(x)
 Tan(x)
 Cot(x)

Even
 Cos(x)
 Sec(x)
Radians

1
C

Radian measure of the
angle at the center of
a unit circle equals the
length of the arc that
the angle cuts from the
unit circle.
Radians
s = 
r
1
s
1
s
 =
r

C
r
Note: Radian measure
is a dimensionless
number
Radians and Degrees
 = s
2 2 r
radian measure
arclength
degree measure
=
=
2
circumference
360°
2 = 360°
 = 180°
Famous Values
Angle
0º = 0
Sin
0 /2 =0
Cos
4 /2 =1
30º = /6
1 /2 =1/2
3 /2
1/ 3
45º = /4
2 /2
2 /2
1
60º = /3
3 /2
1 /2 =1/2
3
90º = /2
4 /2 =1
0 /2 =0
Tan
0
Und
Domain, Range, Period
Domain
Range
Period
sin(x)
(-, )
(-1, 1)
2
cos(x)
(-, )
(-1, 1)
2
tan(x)
x /2,
3/2, ...
(-, )

Finding the Period



Sin (3πx/2 + 4)
Set term that includes x equal to the
period of the trig function
 3πx/2 = 2π
Solve for x
 x = 4/3 = Period
Trig Identities to Know



Pythagorean Identities
 sin2 x + cos2 x = 1
 tan2 x + 1 = sec2 x
 cot2 x + 1 = csc2 x
Double Angle
 sin(2x)=2sin(x)cos(x)
 cos(2x)=cos2(x) – sin2(x)
Square
 sin2(x) = (1 – cos(2x))/2
 cos2(x) = (1 + cos(2x))/2
Creating Inverse Trig
Functions


The trig functions are not 1-1
Restrict their domains
y
= sin(x)
-π/2 ≤ x ≤ π/2
y
= cos(x)
0≤x≤π
y
= tan(x)
-π/2 < x < π/2
The Inverse Trig Functions



y = sin-1 x or y = arcsin(x)
 Domain: [-1, 1]
 Range: [-π/2, π/2]
y = cos-1 x or y = arccos(x)
 Domain: [-1, 1]
 Range: [0, π]
y = tan-1 x or y = arctan(x)
 Domain: (-∞, ∞)
 Range: (-π/2, π/2)
Examples



sin x = 0.455
 x = sin-1 (0.455)
cos x = π/2
 x = cos-1 (π/2)
tan x = 8
 x = tan-1 (8)