Study Guides/Maths/sin3x Formula
Study Guide · Maths

What is the formula for sin(3x)?

In Class 11 Mathematics (Trigonometric Functions), the triple angle formulas are crucial for simplifying complex equations and solving integration problems in Class 12. The formula for sin(3x) is one of the most frequently used identities.

Question (Click to Flip)

What is the formula for cos(3x)?

Answer

The formula for cos(3x) is almost the exact reverse of the sin(3x) formula. cos(3x) = 4 cos³(x) - 3 cos(x).

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Key Facts

This formula is particularly useful in Calculus (Integration). Because integrating sin³(x) directly is difficult, students use this formula to rewrite sin³(x) as (3sin(x) - sin(3x)) / 4, which is very easy to integrate.

The sin(3x) Formula

The formula to expand sin(3x) entirely in terms of sin(x) is:

sin(3x) = 3 sin(x) - 4 sin³(x)

(You can also write this using theta: sin(3θ) = 3sinθ - 4sin³θ).

How is this derived?

The derivation relies on splitting 3x into (2x + x) and using the compound angle formula:

  1. Write sin(3x) as sin(2x + x).
  2. Apply the formula: sin(A+B) = sinA cosB + cosA sinB. So, sin(2x+x) = sin(2x)cos(x) + cos(2x)sin(x).
  3. Now, substitute the double angle formulas:
    • Replace sin(2x) with 2sin(x)cos(x)
    • Replace cos(2x) with (1 - 2sin²(x))
  4. Multiply it out: [2sin(x)cos(x)]cos(x) + [1 - 2sin²(x)]sin(x) = 2sin(x)cos²(x) + sin(x) - 2sin³(x)
  5. Finally, replace cos²(x) with (1 - sin²(x)): = 2sin(x)(1 - sin²(x)) + sin(x) - 2sin³(x) = 2sin(x) - 2sin³(x) + sin(x) - 2sin³(x) = 3sin(x) - 4sin³(x).

Questions and Answers

What is the formula for cos(3x)?+

The formula for cos(3x) is almost the exact reverse of the sin(3x) formula. **cos(3x) = 4 cos³(x) - 3 cos(x)**.

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