Verify the Identity sin(3x)=3sin(x)-4sin(x)^3

Problem

sine 3 x equals 3 sine x minus 4 sine cubed x

Solution

1. Rewrite the argument 3 x as the sum of 2 x and x to prepare for the angle addition formula.

sine 3 x equals the sine of open paren 2 x plus x close paren

2. Apply the sine addition formula the sine of open paren A plus B close paren equals sine A cosine B plus cosine A sine B

the sine of open paren 2 x plus x close paren equals sine 2 x cosine x plus cosine 2 x sine x

3. Substitute the double angle identities sine 2 x equals 2 sine x cosine x and cosine 2 x equals 1 minus 2 sine squared x into the expression.

sine 3 x equals open paren 2 sine x cosine x close paren times cosine x plus open paren 1 minus 2 sine squared x close paren times sine x

4. Simplify the terms by multiplying the trigonometric functions.

sine 3 x equals 2 sine x cosine squared x plus sine x minus 2 sine cubed x

5. Use the Pythagorean identity cosine squared x equals 1 minus sine squared x to express the entire equation in terms of sine.

sine 3 x equals 2 sine x times open paren 1 minus sine squared x close paren plus sine x minus 2 sine cubed x

6. Distribute the 2 sine x through the parentheses.

sine 3 x equals 2 sine x minus 2 sine cubed x plus sine x minus 2 sine cubed x

7. Combine like terms to reach the final form of the identity.

sine 3 x equals 3 sine x minus 4 sine cubed x

Final Answer

sine 3 x equals 3 sine x minus 4 sine cubed x

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