If sin theta +cos theta =1.2, then what is sin^3 theta + cos^3 theta?

Question

Hmm...I don't understand how to proceed. I know I must apply a trig Identity, but which one?

Thanks in advance

Answer 1

sinØ + cosØ = 1.2
square both sides
sin^2 Ø + 2sinØcosØ + cos^2 Ø = 1.44
but sin^2Ø + cos^2Ø = 1
2sinØcosØ = 0.44
sinØcosØ = .22

now using the fact that

(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
we see that
a^3 + b^3 = (a+b)^3 - 3a^2b - 3ab^2
= (a+b)^3 - 3ab(a+b)

so sin^3 Ø + cos^3 Ø
= (sinØ + cosØ)^3 - 3sinØcosØ (sinØ + cosØ)
= 1.2^3 - 3(.22)(1.2)
= 0.936

check:
from 2sinØcosØ = .44
sin 2Ø = .44
2Ø = 26.103...
Ø = 13.0519....
then by calculator: sin^3 13.0519° + cos^3 13.0519°
= .93599...
not bad!

Answer 2

Thanks a lot :)