The trigonometric identity for sin 3x can be derived using the angle addition formulas. Specifically, the formula for sin(a + b) is:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
To find sin(3x), we can rewrite it as sin(2x + x). Thus, applying the angle addition formula gives us:
sin(3x) = sin(2x + x) = sin(2x)cos(x) + cos(2x)sin(x)
Next, we can use the double angle formulas for sin(2x) and cos(2x):
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos²(x) – sin²(x)
Substituting these into our equation, we get:
sin(3x) = (2sin(x)cos(x))cos(x) + (cos²(x) – sin²(x))sin(x)
This simplifies to:
sin(3x) = 2sin(x)cos²(x) + (cos²(x) – sin²(x))sin(x)
Rearranging this further gives:
sin(3x) = 3sin(x) – 4sin³(x)
This is the trigonometric identity for sin 3x. It shows how the sine of three times an angle can be expressed in terms of the sine of the angle itself.