Download UNIT CIRCLE

UNIT CIRCLE
coordinates are
(cos θ, sin θ)
( 0, 1)
►( ,
►(
)
,
►(
π►
(-1, 0)
0 ►(1, 0)
FUNDAMENTALS
cos(- θ ) = cos θ
=
, )
sin(- θ ) = – sin θ
tan θ
=
=
cot θ =
=
sec θ
=
=
csc θ
=
=
sin2 θ + cos2 θ = 1
==
sec2 θ = 1 + tan2 θ
csc2 θ = 1 + cot2 θ
h
θ
y
x
Addition Identities
Double-Angle Identities
sin  2   2 sin  cos
sin( A  B)  sin A cos B  cos A sin B
cos  2   cos2   sin 2 
sin( A  B)  sin A cos B  cos A sin B
 2 cos 2   1
cos( A  B)  cos A cos B  sin A sin B
 1  2sin 2 
cos( A  B)  cos A cos B  sin A sin B
Half-Angle Identities
sin

2

1  cos 
2
cos

2
[Memory tip: “sinus-minus”]

1  cos 
2
Power Reducing Form of Half-Angle Identities
sin 2 A  1 1  cos 2 A
cos2 A  1 1  cos 2 A
2
[Memory tip: “sinus-minus”]
2
Solving Triangles and Areas of Triangles Useful for applications
Law of Sines
sin A sin B sin C


a
b
c
Law of Cosines
B
c
a  b  c  2bc cos A
2
2
2
A
C
b  a  c  2ac cos B
2
2
a
2
b
c  a  b  2ab cos C
2
2
2
Area of triangle given two sides a , b and the included angle 
A
1
ab sin 
2
Heron’s formula for area of triangle given sides a, b, and c
A  s  s  a  s  b  s  c  where s 
abc
2
Inverse Trigonometric Functions


If   sin a , then
If   cos a , then 0    
 
1
1
2
2
If   tan 1 a , then


 
2
2