Download Cofunction and Pythagorean Identities

Cofunction and Pythagorean Identities
PreCalc for AP Prep
Date: ______
I. Identities
A. An equation that is true for all values of all variables it contains.
1. We’ve worked with identities for a long time; we just never described
them that way.
Ex:
a(b + c) = ab + ac
is an identity.
B. Identities get names to help us remember them.
C. Identities work frontwards and backwards.
D. Identities are used to simplify expressions or solve equations.
II. Trigonometric Identities
A. Co-function Relationships:
1. The full list sin   cos(90   )
cos   sin(90   )
tan   cot(90   )
cot   tan(90   )
sec   csc(90   )
csc   sec(90   )
2. A proof of one or two of them.
A
Given right  ABC, show:
c
a. sin A = cos (90 – A)
b
C
a
B
b. sec B = csc (90 – B)
B. The Pythagorean Identites
1. The “full” list:
sin 2 x  cos 2 x  1
1  cot 2 x  csc 2 x
tan 2 x  1  sec 2 x
Notes:
Proof of
sin 2 x  cos 2 x  1
Now that we have informally proved that sin2θ + cos2θ = 1, we can use that to prove the
other Pythagorean Identities. We will do this using the formal t-proof notation from
geometry.
Example #1:
GIVEN: sin2θ + cos2θ = 1
PROVE: tan2θ + 1 = sec2 θ
We need to pay careful attention to the directions that we are given with these problems.
If we are asked to prove an identity, we need reasons. If we are asked to simplify, we
need to work the problem until it is in its simplest form (low number of terms and
trigonometric functions.)
Simplifying and proving identities both require us to NOTICE-REACT-FOLLOW-UP
We can’t help what we notice, and sometimes the correct reactions get some pretty
complex looking results, which is why it is good to know what we hope to follow up
with.
Examples:
Simplify.
1)
csc2x – cot2x
2)
cos tan   cot  
3)
1 – sin2x
4)
sin 2
5)
sec   csc 
1  tan 
6)
sin  cos 

cos  sin 
1
cos  sin 
3
3
 cos 2
4
4
Prove. (Hint: Use the more complicated side of the equation & simplify to prove).
cot x(1  tan 2 x)
11)
csc2 x 
tan x
Classwork:
Simplify
1)
3)
csc x cot x

sin x tan x
(sec2x – 1)(1 – sin2x)
2)
4)
1
 cot 2 x
2
sin x
1
sin 2 
cos  
cos 
Prove:
5)
cot2θ + cos2θ + sin2θ = csc2θ
HW: p. 321 #2, 4, 10, 11, 14,
18, 22, 30, 32, 36