Trigonometric functions
Step one: similar triangles
Two similar triangles have the same set
of angles, and have the properties that
c
a
θ
b
A
a
= ,
B
b
B
b
A
a
= , and
= .
C
c
C
c
Define
C
θ
B
A
cos(✓) =
b
c
and
sin(✓) =
Then let
tan(✓) =
sin(✓)
a
= ,
cos(✓)
b
sec(✓) =
1
c
= ,
sin(✓)
a
csc(✓) =
1
c
= ,
cos(✓)
b
cot(✓) =
1
b
= .
tan(✓)
a
a
.
c
Easy angles:
isosceles right triangle:
equilateral triangle cut in half:
1
√2
1
π/6
1
h
π/3
π/4
1
h=
cos(✓)
sin(✓)
tan(✓)
sec(✓)
p
1/2
1
(1/2)2 =
csc(✓)
p
3/2
cot(✓)
⇡/4
⇡/3
⇡/6
Step two: the unit circle
1
(x , y)
1
θ
x
-1
For 0 < ✓ < ⇡2 ...
x
=x
1
y
sin(✓) = = y
1
cos(✓) =
y
1
-1
Use this idea to extend trig functions to any ✓...
Define
cos(✓) = x
sin(✓) = y ,
there ✓ is defined by...
0  ✓  2⇡
all ✓
1
0
1
✓<0
1
(x , y)
θ
θ
-1
1
-1
1
(x , y)
-1
-1
-1
1
θ
-1
(x , y)
Sidebar: In calculus, radians are king. Where do they come from?
Circumference of a unit circle: 2⇡
Arclength of a wedge with angle ✓:
✓
✓
or
(if in radians)
180 ⇤ 2⇡ (if in degrees)
2⇡ ⇤ 2⇡ = ✓
Reading o↵ of the unit circle
1
2π/3
π/2
3π/4
cos(⇡ ✓) =
π/3
π/6
π
0
-1
cos( ✓) = cos(✓)
sin( ✓) =
sin(✓)
1
cos(2⇡n+✓) = cos(✓)
7π/6
7π/4
4π/3
-1
0
5π/3
cos(✓)
sin(✓)
2⇡
3
⇡
2
⇡
3⇡
4
sin(2⇡n+✓) = sin(✓)
11π/6
x 2 +y 2 = 1 =) cos2 (✓)+sin2 (✓) = 1
3π/2
cos(✓)
sin(✓)
sin(⇡ ✓) = sin(✓)
π/4
5π/6
5π/4
cos(✓)
3⇡
2
5⇡
6
⇡
6
7⇡
6
⇡
4
5⇡
4
⇡
3
4⇡
3
5⇡
3
7⇡
4
11⇡
6
Plotting on the ✓-y axis
Graph of y = cos(✓):
1
-π
A
π
2π
-2π
-1
A = Amplitude = 12 length of the range = 1
T =Period = time to repeat = 2⇡
Graph of y = sin(✓):
1
π
-π
2π
-2π
-1
T
A=Amplitude = 12 length of the range = 1
T =Period = time to repeat = 2⇡
Trig identities to know and love:
Even/odd:
cos( ✓) = cos(✓)
(even)
sin( ✓) =
sin(✓)
(odd)
Pythagorean identity:
cos2 (✓) + sin2 (✓) = 1
Angle addition:
cos(✓ + ) = cos(✓) cos( )
sin(✓) sin( )
sin(✓ + ) = sin(✓) cos( ) + cos(✓) sin( )
(in particular cos(2✓) = cos2 (✓)
sin2 (✓) and sin(2✓) = 2 sin(✓) cos(✓))
Transform the graph of sin(✓) into the graph of 2 sin( 12 ✓ + ⇡/6)
1:
1
π
-π
2π
-2π
-1
3
2
⇣
1
sin 2 ✓
⌘
1
π
-π
!
2π
-2π
-1
-2
-3
3
2
⇣
⌘
1
sin 2 (✓+ ⇡3 )
1
π
-π
!
2π
-2π
-1
-2
-3
3
2
⇣
1
2 sin 2 (✓+ ⇡3 )
⌘
1
π
-π
!
2π
-2π
-1
-2
-3
3
2
⇣
⌘
1
2 sin 2 (✓+ ⇡3 ) 1
1
π
-π
!
2π
-2π
-1
-2
-3
What is the amplitude of 2 sin
1
2✓
+
⇡
6
1?
What is the period?
Other trig functions
y =2 sin(✓)
y =2 cos(✓)
1
1
π
-π
π
-π
-1
-1
-2
-2
sec(✓) = 1/ cos(✓)
csc(✓) = 1/ sin(✓)
2
2
1
1
-π
π
π
-π
-1
-1
-2
-2
Other trig functions
y =2 sin(✓)
y =2 cos(✓)
1
1
π
-π
π
-π
-1
-1
-2
-2
tan(✓) = sin(✓)/ cos(✓)
cot(✓) = cos(✓)/ sin(✓)
2
2
1
1
-π
π
π
-π
-1
-1
-2
-2