Question
Question: Show that \[\sin 3\theta = 3\sin \theta - 4{\sin ^3}\theta \] ?...
Show that sin3θ=3sinθ−4sin3θ ?
Solution
Hint : The question is related to trigonometry which is a sine of triple angle, here we have to prove the given identity by the standard trigonometric formula sine sum and the double angle formula of sine and cosine and on further simplification we get the required proof of the given identity.
Complete step by step solution:
Trigonometric ratios: Some ratios of the sides of a right-angle triangle with respect to its acute angle called trigonometric ratios of the angle.
Let us consider the given trigonometry identity
⇒sin3θ=3sinθ−4sin3θ
Simplify the Left Hand Side (LHS) part until the RHS part becomes the same. Then LHS will be equal to RHS which is written as: LHS= RHS it means the identity was proved.
Consider the LHS
⇒sin3θ ------(1)
The angle can be written in addition form i.e., 3θ=2θ+θ
Then equation (1) becomes
⇒sin(2θ+θ)
As we know the sine sum formula sin(A+B)=sinAcosB+cosAsinB . This formula is also known as trigonometry function for sum of two angles. Where A and B represents the angles.
Here A=2θ and B=θ
Substitute A and B in formula then
⇒sin2θcosθ+cos2θsinθ --------(2)
the double angle formula of cosine: cos(2θ)=1−2sin2θ and
the double angle formula of sine: sin(2θ)=2sinθcosθ
substitute double angle formulas in equation (2), then
⇒sin(2θ)cosθ+cos(2θ)sinθ
⇒(2sinθcosθ)cosθ+(1−2sin2θ)sinθ
⇒2sinθcos2θ+sinθ−2sin3θ ------(3)
Use the fundamental identity of trigonometry sin2θ+cos2θ=1 ⇒cos2θ=1−sin2θ .
Equation (3) becomes
⇒2sinθ(1−sin2θ)+sinθ−2sin3θ
On multiplying 2sinθ
⇒2sinθ−2sin3θ+sinθ−2sin3θ
On simplification, we get
⇒3sinθ−4sin3θ
= RHS
∴LHS=RHS
Hence showed
sin3θ=3sinθ−4sin3θ
So, the correct answer is “ sin3θ=3sinθ−4sin3θ ”.
Note : The question is involving the trigonometry terms. Here we must know about the trigonometry functions of sum of two angles and double angle trigonometry ratios. By using the above formulas, we are going to simplify the given trigonometric function. While simplifying we should take care of signs.