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7-3 Identities Day 1
Identities
•Identities are like _______________________
•We will work on one side only. Usually the
HARDER looking side.
•If there are fractions ____________________
•______________if it helps!
Need to remember:
tanx  ________
sin2 x  cos2 x  1
sec x  ________
1  tan x  sec x
csc x  ________
1  cot x  csc x
cot x  ____________
2
2
2
2
Examples
cos x  sin x  2cos x  1
2
2
2
Need some hints?
1. Choose harder side. Tossup? Try the
left side.
2. Fractions? ___________________.
3. __________, if it helps.
4. One side in many functions/ other side in
one? Work on the ______________ side
and get it into the only function seen on
the other side.
sec 2 x
2
1.

csc
x
2
sec x  1
1  tanθ
2.
 csc θ  sec θ
sinθ
1  sinθ
cosθ
3.

cosθ
1  sinθ
tanθsinθ
5.
 cosθ
2
sec θ  1
7-4 Formulas
Angle Addition/Subtraction
Lets try an experiment?
What is cos  30  60
?
__________
Is it equal to cos30  cos60 ?
This demonstrates that trigonometric functions
____________________________
Imagine
(4  9)
The Formulas
Based on Geometry and distance Formula
cos  A  B  ________________
cos  A  B  ________________
sin  A  B  ________________
sin  A  B  ________________
The Formulas
Based on Geometry and distance Formula
tan  A  B  _____________
tan  A  B  _______________
What do these formulas do?
Essentially give you the option of
a) finding the sin/cos of an angle that can
be made up of two angles you know.
While you don’t know the cos 75 °, you
know that 75° is made up of two angles
that you know.
b) Going backwards, and “compressing” an
expanded formula into a form you can
solve.
Examples
Find the following
1. cos75
Prove sin 90  1 by angle addition
formulas.
cos202 cos22  sin202 sin22
Example
7.
Given cos A  3
Find
5
8
,
sinB

17
and
0 A 

3
, B
2
2
sin(A  B), cos(A  B)
Class Work
Find the following
4.
sin15
5.
cos225
6.
tan105
7-5 More Formulas
Double Angle
Half Angle
Lets derive some formulas:
What is cos  A  A  ?
cos 2A  cos A  sin A
2
2
What is sin  A  A 
sin2A  2sin A cos A
What is tan  A  A 
2 tan A
tan 2A 
2
1  tan A
Some other versions
cos2A  2cos A  1  1  2sin A
2
2
Example
Evaluate sin 60 using double angles.
If
find
5
sin  
and  is in Q II
13
cos2
5
13

-12
Find b
4
b
Θ
Θ
2
Half Angle Formulas
A
1  cos A
sin  
2
2
A
1  cos A
cos  
2
2
A
sin A
1  cos A
tan 

2 1  cos A
sin A
Examples
1. Evaluate sin 15 using half angle.
5

2.If sin    and  is in Q IV find cos
13
2
1
Find cos 157 
2