To find the formula of cos 3A, we can use the cosine angle addition formulas.

• We start with cos(3A) = cos(2A + A).
• Using the cosine addition formula, we have cos(2A + A) = cos(2A)cos(A) - sin(2A)sin(A).
• Next, we use the double angle formulas: cos(2A) = 2cos²(A) - 1 and sin(2A) = 2sin(A)cos(A).
• Substituting these into our equation gives us: cos(3A) = (2cos²(A) - 1)cos(A) - (2sin(A)cos(A))sin(A).
• Simplifying this further leads to: cos(3A) = 4cos³(A) - 3cos(A).

To derive the formula for cos(3A), we start by expressing it using the cosine addition formula. We can write:

Step 1: Using the Cosine Addition Formula

We know that:

cos(3A) = cos(2A + A)

Applying the cosine addition formula, we have:

cos(2A + A) = cos(2A)cos(A) - sin(2A)sin(A)

Step 2: Applying Double Angle Formulas

Next, we use the double angle formulas:

• cos(2A) = 2cos²(A) - 1
• sin(2A) = 2sin(A)cos(A)

Step 3: Substituting Values

Substituting these into our equation gives:

cos(3A) = (2cos²(A) - 1)cos(A) - (2sin(A)cos(A))sin(A)

Step 4: Simplifying the Expression

Now, we simplify this expression:

First, expand the terms:

cos(3A) = 2cos³(A) - cos(A) - 2sin²(A)cos(A)

Since sin²(A) = 1 - cos²(A), we can substitute:

cos(3A) = 2cos³(A) - cos(A) - 2(1 - cos²(A))cos(A)

After simplifying, we arrive at:

cos(3A) = 4cos³(A) - 3cos(A)

This final formula, cos(3A) = 4cos³(A) - 3cos(A), is useful in various applications in trigonometry.