11x1 t04 03 pythagorean trig identities (2011)
TRANSCRIPT
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