Simplify sin(2arcsin(1/2)-arctan(( square root of 3)/3))

$\sin (2 \arcsin (\frac{1}{2}) - \arctan (\frac{\sqrt{3}}{3}))$

Simplify each term.

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$\sin (2 \frac{π}{6} - \arctan (\frac{\sqrt{3}}{3}))$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$\sin (2 \frac{π}{2 (3)} - \arctan (\frac{\sqrt{3}}{3}))$

Cancel the common factor.

$\sin (2 \frac{π}{2 \cdot 3} - \arctan (\frac{\sqrt{3}}{3}))$

Rewrite the expression.

$\sin (\frac{π}{3} - \arctan (\frac{\sqrt{3}}{3}))$

The exact value of $\arctan (\frac{\sqrt{3}}{3})$ is $\frac{π}{6}$ .

$\sin (\frac{π}{3} - \frac{π}{6})$

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sin (\frac{π}{3} \cdot \frac{2}{2} - \frac{π}{6})$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$\sin (\frac{π \cdot 2}{3 \cdot 2} - \frac{π}{6})$

Multiply $3$ by $2$ .

$\sin (\frac{π \cdot 2}{6} - \frac{π}{6})$

Combine the numerators over the common denominator.

$\sin (\frac{π \cdot 2 - π}{6})$

Simplify the numerator.

Move $2$ to the left of $π$ .

$\sin (\frac{2 \cdot π - π}{6})$

Subtract $π$ from $2 π$ .

$\sin (\frac{π}{6})$

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{1}{2}$

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$