3.8 Fundamental Identities

Ex 1) Use definitions to prove:

–A trig identitiy is a trig equation that is always true

–We can prove an identity using the definitions of trig functions

(they use x, y, and r)

sin cot cos

cosy x yx x

r y ry r

We also have the Pythagorean Identities

2 21 tan sec

2 2cos sin 1

2 21 cot csc

“I tan in a second”

(get by ÷ by cos2θ)

“I cotan in a cosecond”

(get by ÷ by sin2θ)

tan 33

tan 3 33

We can prove identities (using θ, ϕ, β, etc) or verify the identity using specific values.

Ex 2) Use exact values to verify the identity for the given θ

a) tan( ) tan ;3

1

260°

3

2

1

LHS:

RHS:

2

2

3 342

1 1 3 1 41 1

3 3 3 33

1 1 4 41

3 3

Ex 2) Use exact values to verify the identity for the given θ

b) 2 21 tan sec ; 150

150°1

230°

3

2

1

LHS:

RHS:

Other Identities to use:

sin costan cot

cos sin

sin( ) sin

csc( ) csc

Ratio:

Reciprocal:

Pythagorean Identities: (already mentioned)

Odd/ Even:

1 1 1csc sec cot

sin cos tan

tan( ) tan

cot( ) cot

cos( ) cos

sec( ) sec

2tan sintan

tan cos

(try ratio & reciprocal)

b)

Ex 3) Simplify by writing in terms of sine & cosine

a)

Pythag (1 + tan2θ = sec2θ)

odd/even

cot sec

cos 1 1

sin cos sin

2sec 1

tan( )

1

Homework#308 Pg 169 #1–45 odd

Hints for HW Make sure calculator is in correct MODE Draw those reference triangle pictures!