What is the formula for sin(3x) and how can it be derived?

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The Formula for sin(3x)

The formula for sinus of a triple angle, specifically sin(3x), can be derived using the angle addition formulas and identities from trigonometry. The formula is given by:

sin(3x) = 3sin(x) – 4sin3(x)

This means that the sine of three times an angle x can be expressed in terms of the sine of the angle x itself.

Derivation of the Formula

To derive this formula, we can start with the angle addition formula for sine:

sin(a + b) = sin(a) cos(b) + cos(a) sin(b)

We can apply this formula by setting a = 2x and b = x, leading us to:

sin(3x) = sin(2x + x) = sin(2x) cos(x) + cos(2x) sin(x)

Expanding sin(2x) and cos(2x)

Next, we apply the double angle formulas:

  • sin(2x) = 2sin(x)cos(x)
  • cos(2x) = cos2(x) – sin2(x)

Substituting Back

Substituting these back into the equation gives us:

sin(3x) = (2sin(x)cos(x))cos(x) + (cos2(x) – sin2(x))sin(x)

Simplifying

Now we simplify this expression step by step:

  1. sin(3x) = 2sin(x)cos2(x) + cos2(x)sin(x) – sin3(x)
  2. Combine like terms: sin(3x) = (2cos2(x) + cos2(x) – sin2(x))sin(x)
  3. This further simplifies to: sin(3x) = (3cos2(x) – sin2(x))sin(x)

Utilizing Pythagorean identity, where cos2(x) = 1 – sin2(x):

  1. Replace cos2(x):

    sin(3x) = (3(1 – sin2(x)) – sin2(x))sin(x)
  2. Distributing gives:

    sin(3x) = (3 – 4sin2(x))sin(x)

Thus, we arrive at the final formula:

sin(3x) = 3sin(x) – 4sin3(x)

This formula is particularly useful in various applications, including solving trigonometric equations and analyzing wave functions.

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