Question
If sinα+sinβ+sinγ=0= cosα+cosβ+cosγ,
value of sin2α+sin2β+sin2γ
value of sin2α+sin2β+sin2γ
Answer: Option B
:
B
cosα+cosβ+cosγ = 0............(i)
sinα+sinβ+sinγ =0...........(ii)
Let a=cosα+isinα, b=cosβ+isinβ
c=cosγ+isinγ
⇒ a+b+c=0[By (i) and (ii)]....(iii)
1a+1b+1c
=cosα-isinα+cosβ-isinβ+cosγ-isinγ
⇒ ab+bc+ca=0 ......(iv) [by (i) and (ii)]
Squaring both sides of equation (iii),
we get a2+b2+c2+2ab+2bc+2ca=0
or a2+b2+c2=0 [by (iv)]
[cosα+isinα]2 + [cosβ+isinβ]2 + [cosγ+isinγ]2 = 0
⇒ (cos2α+cos2β+cos2γ) + i(sin2α+sin2β+sin2γ)=0
Separation of real and imaginary part,
⇒ cos2α+cos2β+cos2γ=0 .........(v)
And sin2α+sin2β+sin2γ =0 .........(iv)
⇒ 1-2sin2α+1-2sin2β+1-2sin2γ=0 [By eq.(v)]
Therefore, sin2α+sin2β+sin2γ= 32
Was this answer helpful ?
:
B
cosα+cosβ+cosγ = 0............(i)
sinα+sinβ+sinγ =0...........(ii)
Let a=cosα+isinα, b=cosβ+isinβ
c=cosγ+isinγ
⇒ a+b+c=0[By (i) and (ii)]....(iii)
1a+1b+1c
=cosα-isinα+cosβ-isinβ+cosγ-isinγ
⇒ ab+bc+ca=0 ......(iv) [by (i) and (ii)]
Squaring both sides of equation (iii),
we get a2+b2+c2+2ab+2bc+2ca=0
or a2+b2+c2=0 [by (iv)]
[cosα+isinα]2 + [cosβ+isinβ]2 + [cosγ+isinγ]2 = 0
⇒ (cos2α+cos2β+cos2γ) + i(sin2α+sin2β+sin2γ)=0
Separation of real and imaginary part,
⇒ cos2α+cos2β+cos2γ=0 .........(v)
And sin2α+sin2β+sin2γ =0 .........(iv)
⇒ 1-2sin2α+1-2sin2β+1-2sin2γ=0 [By eq.(v)]
Therefore, sin2α+sin2β+sin2γ= 32
Was this answer helpful ?
More Questions on This Topic :
Question 1. The value of i1+3+5+......+(2n+1) is....
Question 4. The value of i1+3+5+......+(2n+1) is....
Question 8. The square root of 3 - 4i is ....
Submit Solution