Pythagorean Trig Identities

Pythagorean Trig Identities

y

x

1

xy

Pythagorean Trig Identities

y

x

2 2 1x y

1

xy

Pythagorean Trig Identities

y

x

2 2 1x y

sin1sin

y

y

1

xy

Pythagorean Trig Identities

y

x

2 2 1x y

sin1sin

y

y

cos1cos

x

x

1

xy

Pythagorean Trig Identities

y

x

2 2 1x y

sin1sin

y

y

cos1cos

x

x

2 2sin cos 1

1

xy

Pythagorean Trig Identities

y

x

2 2 1x y

sin1sin

y

y

cos1cos

x

x

2 2sin cos 1

2Divide by sin

1

xy

Pythagorean Trig Identities

y

x

2 2 1x y

sin1sin

y

y

cos1cos

x

x

2 2sin cos 1

2Divide by sin 2 2

2 2 2sin cos 1sin sin sin

1

xy

Pythagorean Trig Identities

y

x

2 2 1x y

sin1sin

y

y

cos1cos

x

x

2 2sin cos 1

2Divide by sin 2 2

2 2 2sin cos 1sin sin sin

2 21 cot cosec

1

xy

Pythagorean Trig Identities

y

x

2 2 1x y

sin1sin

y

y

cos1cos

x

x

2 2sin cos 1

2Divide by sin 2 2

2 2 2sin cos 1sin sin sin

2 21 cot cosec

1

xy

2Divide by cos

Pythagorean Trig Identities

y

x

2 2 1x y

sin1sin

y

y

cos1cos

x

x

2 2sin cos 1

2Divide by sin 2 2

2 2 2sin cos 1sin sin sin

2 21 cot cosec

1

xy

2Divide by cos 2 2

2 2 2sin cos 1cos cos cos

Pythagorean Trig Identities

y

x

2 2 1x y

sin1sin

y

y

cos1cos

x

x

2 2sin cos 1

2Divide by sin 2 2

2 2 2sin cos 1sin sin sin

2 21 cot cosec

1

xy

2Divide by cos 2 2

2 2 2sin cos 1cos cos cos

2 2tan 1 sec

Pythagorean Trig Identities

y

x

2 2 1x y

sin1sin

y

y

cos1cos

x

x

2 2sin cos 1

2Divide by sin 2 2

2 2 2sin cos 1sin sin sin

2 21 cot cosec

1

xy

2Divide by cos 2 2

2 2 2sin cos 1cos cos cos

2 2tan 1 sec

2 2sin cos 1 2 21 tan sec 2 21 cot cosec

2 2sin cos 1 2 21 tan sec 2 21 cot cosec

In addition: sin tancos

2 2sin cos 1 2 21 tan sec 2 21 cot cosec

In addition: sin tancos

3e.g. ( ) If cos and tan is negative, find sin4

i

2 2sin cos 1 2 21 tan sec 2 21 cot cosec

In addition: sin tancos

3e.g. ( ) If cos and tan is negative, find sin4

i

3 4

2 2sin cos 1 2 21 tan sec 2 21 cot cosec

In addition: sin tancos

3e.g. ( ) If cos and tan is negative, find sin4

i

3 4

7

2 2sin cos 1 2 21 tan sec 2 21 cot cosec

In addition: sin tancos

3e.g. ( ) If cos and tan is negative, find sin4

i

3 4

7

4th quadrant

2 2sin cos 1 2 21 tan sec 2 21 cot cosec

In addition: sin tancos

3e.g. ( ) If cos and tan is negative, find sin4

i

3 4

7

4th quadrant

7sin4

(ii) Simplify;2) 1 cosa

(ii) Simplify;2) 1 cosa

21 1 sin

(ii) Simplify;2) 1 cosa

21 1 sin 2

2

1 1 sinsin

(ii) Simplify;2) 1 cosa

21 1 sin 2

2

1 1 sinsin

2 2) sec tanb A A

(ii) Simplify;2) 1 cosa

21 1 sin 2

2

1 1 sinsin

2 2) sec tanb A A2 21 tan tanA A

1

(ii) Simplify;2) 1 cosa

21 1 sin 2

2

1 1 sinsin

2 2) sec tanb A A2 21 tan tanA A

1

) tan cosc

(ii) Simplify;2) 1 cosa

21 1 sin 2

2

1 1 sinsin

2 2) sec tanb A A2 21 tan tanA A

1

) tan cosc sin coscos 1

sin

(ii) Simplify;2) 1 cosa

21 1 sin 2

2

1 1 sinsin

2 2) sec tanb A A2 21 tan tanA A

1

) tan cosc sin coscos 1

sin

22sec 2 tan Prove

4 tan 2iii

(ii) Simplify;2) 1 cosa

21 1 sin 2

2

1 1 sinsin

2 2) sec tanb A A2 21 tan tanA A

1

) tan cosc sin coscos 1

sin

22sec 2 tan Prove

4 tan 2iii

22sec 24 tan

(ii) Simplify;2) 1 cosa

21 1 sin 2

2

1 1 sinsin

2 2) sec tanb A A2 21 tan tanA A

1

) tan cosc sin coscos 1

sin

22sec 2 tan Prove

4 tan 2iii

22sec 24 tan

22 1 tan 2

4 tan

(ii) Simplify;2) 1 cosa

21 1 sin 2

2

1 1 sinsin

2 2) sec tanb A A2 21 tan tanA A

1

) tan cosc sin coscos 1

sin

22sec 2 tan Prove

4 tan 2iii

22sec 24 tan

22 1 tan 2

4 tan

22 2 tan 24 tan

(ii) Simplify;2) 1 cosa

21 1 sin 2

2

1 1 sinsin

2 2) sec tanb A A2 21 tan tanA A

1

) tan cosc sin coscos 1

sin

22sec 2 tan Prove

4 tan 2iii

22sec 24 tan

22 1 tan 2

4 tan

22 2 tan 24 tan

22 tan4 tan

(ii) Simplify;2) 1 cosa

21 1 sin 2

2

1 1 sinsin

2 2) sec tanb A A2 21 tan tanA A

1

) tan cosc sin coscos 1

sin

22sec 2 tan Prove

4 tan 2iii

22sec 24 tan

22 1 tan 2

4 tan

22 2 tan 24 tan

22 tan4 tan

tan2

(ii) Simplify;2) 1 cosa

21 1 sin 2

2

1 1 sinsin

2 2) sec tanb A A2 21 tan tanA A

1

) tan cosc sin coscos 1

sin

22sec 2 tan Prove

4 tan 2iii

22sec 24 tan

22 1 tan 2

4 tan

22 2 tan 24 tan

22 tan4 tan

tan2

when proving trig identitiesyou can;

1. start with LHS and prove RHS2. start with RHS and prove LHS

3. work on LHS and RHS independently and show they

equal the same thing

NEVER SOLVE LIKE AN EQUATION

Exercise 4E; 1a, 2b, 3ac, 4bd, 5, 7, 9*

Exercise 4F; 3a, 4b, 5c, 6ac, 7bd, 8ac, 10ad,11acegi, 12bdfhj, 13ac, 14bd, 15ac,

16acegik, 17*a