Question

What is the maximum value of

[9sinθ+2sin(θ+π/3)]^2?

Answer 1

let y = [9sinθ+2sin(θ+π/3)]^2

dy/dØ = (9sinθ+2sin(θ+π/3)) (9cosØ + 2cos(Ø+π/3)
= 0 for a max/min

so (9sinθ+2sin(θ+π/3)) =0 OR (9cosØ + 2cos(Ø+π/3)) = 0

(Where are you getting these questions from, they are tediously long....)

Case1:
9sinØ + 2sin(Ø+π/3) = 0
9sinØ + 2(sinØcosπ/3 + cosØsinπ/3) = 0
9sinØ + 2( (1/2)sinØ + (√3/2)cosØ ) = 0
9sinØ + sinØ + √3cosØ = 0
10sinØ = -√3cosØ
sinØ/cosØ = -√3/2
tanØ = -√3/2
Ø could be in II or IV
in II sinØ = √3/√103 and cosØ = -10/√103

but remember I showed that
9sinØ + 2sin(Ø+π/3)
= 10sinØ + √3cosØ

then y = [9sinØ + 2sin(Ø+π/3)]^2
= (10sinØ + √3cosØ)^2
= (10√3/√103 + √3(-10/√103))^2
= 0 , how about that ?

in IV sinØ = -√3/√103 and cosØ = 10/√103
and (10sinØ + √3cosØ)^2
= 0 as well

HALF WAY DONE

Case 2:
9cosØ + 2cos(Ø+π/3) = 0
9cosØ + 2(cosØcos π/3 - sinØsin π/3) = 0
9cosØ + 2( (1/2)cosØ - (√3/2)sinØ ) = 0
9cosØ + cosØ - √3sinØ = 0
10cosØ = √3sinØ
10/√3 = sinØ/cosØ
tanØ = 10/√3
Ø could be in I or III

in I , sinØ = 10/√103 , cosØ = √3/√103

again, recall that
9sinθ+2sin(θ+π/3) = 10sinØ +√3cosØ

y = (10(10/√103) + √3(√3/√103)^2
= (100/√103 + 3/√103)^2
= (103/√103)^2
= 10000/103 = appr 97.09

in III , sinØ = -√3/√103 , cosØ = -10/√103

y = same as above, except
= (-103/√103)^2 = 10000/103 = appr 97.09

so the max is 97.09 , and the minimum is 0

Wolfram appears to confirm my answer
http://www.wolframalpha.com/input/?i=plot+%5B9sinθ%2B2sin%28θ%2Bπ%2F3%29%5D%5E2

Answer 2

Thank you very much! You're good! :)