How do you prove sin 3θ = 3sin θ - 4sin³θ ?

To prove the identity sin 3θ = 3sin θ - 4sin³θ, we can use the angle addition formulas and some algebraic manipulation. This proof involves breaking down the expression for sin 3θ using known trigonometric identities. Let's walk through the steps together.

Step 1: Use the Angle Addition Formula

The sine of a triple angle can be expressed using the angle addition formula. We know that:

• sin(A + B) = sin A cos B + cos A sin B

For sin 3θ, we can write it as sin(2θ + θ). Applying the angle addition formula gives us:

sin 3θ = sin(2θ + θ) = sin 2θ cos θ + cos 2θ sin θ.

Step 2: Substitute for sin 2θ and cos 2θ

Next, we need to express sin 2θ and cos 2θ in terms of sin θ. The double angle formulas are:

• sin 2θ = 2sin θ cos θ
• cos 2θ = 1 - 2sin²θ

Substituting these into our expression for sin 3θ gives:

sin 3θ = (2sin θ cos θ) cos θ + (1 - 2sin²θ) sin θ.

Step 3: Simplify the Expression

Now, let's simplify the expression step by step:

sin 3θ = 2sin θ cos²θ + sin θ - 2sin³θ.

Recall that cos²θ can be rewritten using the Pythagorean identity:

cos²θ = 1 - sin²θ.

Substituting this into our equation gives:

sin 3θ = 2sin θ (1 - sin²θ) + sin θ - 2sin³θ.

Step 4: Expand and Combine Like Terms

Expanding the equation results in:

sin 3θ = 2sin θ - 2sin³θ + sin θ - 2sin³θ.

Combining like terms yields:

sin 3θ = 3sin θ - 4sin³θ.

Final Result

Thus, we have successfully shown that:

sin 3θ = 3sin θ - 4sin³θ.

This identity is now proven using trigonometric identities and algebraic manipulation. It illustrates how the sine function behaves under multiple angles and is a useful result in various applications of trigonometry.