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

Question

asked Dec 25, 2021 143k views

1 Answer

KhaoulaAtallah · Answer 1 · 2021-12-28T23:07:06+0000

Answer:

$\displaystyle (118)/(125)$

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:

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

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

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

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

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

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

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