Download Cofunction and Pythagorean Identities

Cofunction and Pythagorean Identities
Pre-Calc. for AP Prep.
Date: ______
Goal:
I. Trigonometric Identities – Used to simplify or expand expressions to make solving
equations easier. (Think factoring, making a common denominator or combining like
term)
A. Using identities we can change the trigonometric functions in an expression,
the number of terms in an expression or the angles in an expression.
B. Classifying trigonometric identities by their groups helps us talk about them
and can help us remember them.
1. “In terms of sine and cosine”
tan x 
cot x 
sec x 
csc x 
2. “Reciprocal identities”
sin x 
1
cos x 
1
tan x 
1
cot x 
1
a. Spin-offs of those above use the same parts but in different roles.
sin x csc x  _______
cot x  ________  1
C. Cofunction Identities – These are the first ones in which the angle in the expression
changes.

Cofunction Relationships:
sin   cos(90   )
cos   sin(90   )
tan   cot(90   )
cot   tan(90   )
sec   csc(90   )
csc   sec(90   )
D. The Pythagorean Identities
From the unit circle:
Notes:
Now that we have informally proven 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.
GIVEN: sin2θ + cos2θ = 1
PROVE: tan2θ + 1 = sec2 θ
Example #2:
GIVEN: sin2θ + cos2θ = 1
PROVE: 1 + cot2 θ = csc2 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 MUST use a t-proof. 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, no fractions (if possible).)
Skills that might be useful to simplify identities:
a.
write everything in terms of sines and cosines
b.
simplify complex fractions
c.
combine fractions using common denominators
d.
factor
e.
use conjugates
f.
pull apart fractions (rare but sometimes necessary)
Examples:
Simplify.
1)
secxsinx
2)
csc2x – cot2x
3)
1 – sin2x
4)
sin 2
5)
csc x cot x

sin x tan x
6)
1
 cot 2 x
2
sin x
7)
(sec2x – 1)(1 – sin2x)
8)
sec   csc 
1  tan 
3
3
 cos 2
4
4
9)
sin  cos 

cos  sin 
1
cos  sin 
10)
1
cos  
sin 2 
cos 
Prove. (Hint: Use the more complicated side of the equation & simplify to prove).
11)
cot x(1  tan 2 x)
csc x 
tan x
12)
cot2θ + cos2θ + sin2θ = csc2θ
2
HW: p. 321 2, 4, 10, 11, 14, 18, 22, 30, 32, 36