Question

What is the value of sin {3sin^-1 (2/5)} ?​

Answer 1

`\displaystyle (118)/(125)`

Explanation:

Trigonometry

Calculate

`\sin (3\sin^(-1)(2/5))`

We use the formula for the sine of triple angle:

`\sin 3x=3\sin x-4\sin^3 x`

And recall:

`\sin \sin^(-1)x=x`

For this problem, we set:

`x=\sin^(-1)(2/5)`

Thus:

`\sin 3x=3\sin (\sin^(-1)(2/5))-4\sin^3 (\sin^(-1)(2/5))`

`\displaystyle \sin 3x=3*(2)/(5)-4\left((2)/(5)\right)^3`

`\displaystyle \sin 3x=(6)/(5)-4*(8)/(125)`

`\displaystyle \sin 3x=(6)/(5)-(32)/(125)`

`\displaystyle \sin 3x=(6*25)/(125)-(32)/(125)`

`\displaystyle \sin 3x=(150-32)/(125)`

`\mathbf{\displaystyle \sin 3x=(118)/(125)}`