Dear friends, Please read our latest blog post for an important announcement about the website. ❤, The Socratic Team

# How do you simplify sin (2 * arcsin (x))?

Then teach the underlying concepts
Don't copy without citing sources
preview
?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

92
Nov 8, 2016

The answer is $= 2 x \sqrt{1 - {x}^{2}}$

#### Explanation:

Let $y = \arcsin x$, then $x = \sin y$

$\sin \left(2 \arcsin x\right) = \sin 2 y = 2 \sin y \cos y$

${\cos}^{2} y + {\sin}^{2} y = 1$

${\cos}^{2} y = 1 - {x}^{2}$$\implies$$\cos y = \sqrt{1 - {x}^{2}}$

$\therefore \sin \left(2 \arcsin x\right) = 2 x \sqrt{1 - {x}^{2}}$

Then teach the underlying concepts
Don't copy without citing sources
preview
?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

1
Dean R. Share
Jun 15, 2018

If we interpret $\arcsin a$ as all the solutions to $\sin x = a$ then

$\sin \left(2 \arcsin x\right) = 2 \left(\sin \arcsin x\right) \left(\cos \arcsin x\right) = 2 x \cos \arcsin \left(\frac{x}{1}\right) = \pm 2 x \sqrt{1 - {x}^{2}}$

#### Explanation:

$\arcsin \left(\frac{x}{1}\right)$ refers to a right triangle, opposite $x$, hypotenuse $1$ so adjacent $\sqrt{1 - {x}^{2}}$. The sign is ambiguous so we prepend $\pm$.

• 42 minutes ago
• 43 minutes ago
• 46 minutes ago
• 46 minutes ago
• 2 minutes ago
• 6 minutes ago
• 14 minutes ago
• 35 minutes ago
• 36 minutes ago
• 39 minutes ago
• 42 minutes ago
• 43 minutes ago
• 46 minutes ago
• 46 minutes ago
• Che
• Phy
• Org
• Ast
• Psy
• Ana
• Pre
• Ast
• Alg
• Eng
• Pre
• Sta
• Tri
• Geo
• Cal
• Wor
• Env
• Ear
• Bio
• Soc
• U.S