Show that [ costheta + isintheta]^ 3= cos3theta + isin3theta
Answers
Answer:
Cos 3θ + i Sin3θ
Step-by-step explanation:
To prove --->
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(Cosθ + i Sinθ )³ = Cos3θ + i Sin3θ
Proof --->We have an identity
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(a + b )³ = a³ + b³ + 3 a b ( a + b )
Now
LHS =
( Cosθ + i Sinθ )³ =( Cosθ)³ + (i Sinθ )³ +3
Cosθ (i Sinθ ) ( Cosθ + i Sinθ )
= Cos³θ + i³ Sin³θ +3i Cos²θ Sinθ+3i²
Sin²θ Cosθ
=Cos³θ + i² i Sin³θ +3i Cos²θ Sinθ +3(-1)
Sin²θ Cosθ
=Cos³θ - 3 Sin²θ Cosθ + (-1) i Sin³θ +3 i
Cos²θ Sinθ
=Cos³θ -3(1 - Cos²θ ) Cosθ -i Sin³θ +3i
Cos²θ Sinθ
=Cos³θ -3Cosθ +3 Cos³θ - i (Sin³θ -3
Cos²θ Sinθ)
=4Cos³θ-3Cosθ - i{Sin³θ -3 (1-Sin²θ)Sinθ}
= Cos3θ - i (Sin³θ -3 Sinθ + 3 Sin³θ)
= Cos3θ - i (4 Sin³θ - 3Sinθ)
= Cos3θ + i (3 Sinθ - 4 Sin³θ)
=Cos 3θ + i Sin 3θ = RHS