Trig Functions
Definitions
C
θ
B
A
A sin θ =
C csc θ =
C
A cos θ =
B
C
C sec θ =
B tan θ =
A
B cot θ =
B
A
Radians
For use in calculus, angles are best measured in units called radians. By definition, an arc of length θ on a circle of radius one subtends an angle of θ radians at the center of the circle. Because the circumference of a circle of radius one is 2 π , we have
θ
θ
2 π radians = 360
◦
π
3 radians = 60
◦
π radians = 180
◦
π
2 radians = 90
◦
π
4 radians = 45
◦
π
6 radians = 30
◦
Special Triangles
From the triangles above, we have
θ
0
◦
= 0 rad
30
◦
= π
6
45
◦
=
π
4 rad rad
60
◦
=
π
3 rad
90
◦
=
π
2 rad
180
◦
= π rad
√
2
45
◦
1
1
2
60
◦
1
30
√
◦
3
2
60
◦
1 sin θ cos θ tan θ csc θ sec θ cot θ
0
1
2
√
2
√
3
2
1
0
1
√
3
2
√
2
1
2
0
−
1
0
√
3
1
√
3
0
2
√
1
2
√
3
1
√
√
2
−
3
2
1
√
3
1
√
3
0
The empty boxes mean that the trig function is undefined (i.e.
±∞ ) for that angle.
1

Trig Identities – Elementary
The following identities are all immediate consequences of the definitions at the top of the previous page csc θ =
1 sin θ sec θ =
1 cos θ tan θ = sin θ cos θ cot θ
Because 2 π radians is 360
◦
, the angles θ and θ + 2 π are really the same, so
=
1 tan θ
= cos θ sin θ sin( θ + 2 π ) = sin θ cos( θ + 2 π ) = cos θ
The following trig identities are consequences of the figure to their right.
sin(
−
θ ) =
− sin
2 sin(
θ + cos
θ )
2
θ cos(
= 1
−
θ ) = cos( θ )
(cos θ, sin θ )
C = 1
A = sin θ
θ
−
θ B = cos θ
(cos θ,
− sin θ )
The following trig identities are consequences of the figure to their left.
1
π
2
−
θ sin θ sin
π
2
−
θ = cos θ cos
π
2
−
θ = sin θ
θ cos θ
Trig Identities – Addition Formulae
The following trig identities are derived in the handout entitled “Trig Identities – Cosine law and
Addition Formulae” sin( x + y ) = sin x cos y + cos x sin y sin( x
− y ) = sin x cos y
− cos x sin y cos( x + y ) = cos x cos y
− sin x sin y cos( x
− y ) = cos x cos y + sin x sin y
Setting y = x gives sin(2 x ) = 2 sin x cos x cos(2 x ) = cos
2 x − sin
2 x
= 2 cos
2 x
−
1 since sin
2 x = 1
− cos
2 x
= 1
−
2 sin
2 x since cos
2 x = 1
− sin
2 x
Solving cos(2 x ) = 2 cos 2 x
−
1 for cos 2 x and cos(2 x ) = 1
−
2 sin
2 x for sin
2 x gives cos
2 x =
1+cos(2 x )
2 sin
2 x =
1
− cos(2 x )
2
2