Find the Inverse y=1/2*sin(x)

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Trigonometry

$y = \frac{1}{2} \cdot \sin (x)$

Interchange the variables.

$x = \frac{1}{2} \cdot \sin (y)$

Solve for $y$ .

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Rewrite the equation as $\frac{1}{2} \cdot \sin (y) = x$ .

$\frac{1}{2} \cdot \sin (y) = x$

Multiply each term by $2$ and simplify.

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Multiply each term in $\frac{1}{2} \cdot \sin (y) = x$ by $2$ .

$(\frac{1}{2} \cdot \sin (y)) \cdot 2 = x \cdot 2$

Simplify $(\frac{1}{2} \cdot \sin (y)) \cdot 2$ .

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Combine $\frac{1}{2}$ and $\sin (y)$ .

$\frac{\sin (y)}{2} \cdot 2 = x \cdot 2$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{\sin (y)}{2} \cdot 2 = x \cdot 2$

Rewrite the expression.

$\sin (y) = x \cdot 2$

Move $2$ to the left of $x$ .

$\sin (y) = 2 x$

Take the inverse sine of both sides of the equation to extract $y$ from inside the sine.

$y = \arcsin (2 x)$

Solve for $y$ and replace with $f^{- 1} (x)$ .

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Replace the $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \arcsin (2 x)$

Set up the composite result function.

$g (f (x))$

Evaluate $g (f (x))$ by substituting in the value of $f$ into $g$ .

$g (\frac{1}{2} \cdot \sin (x)) = \arcsin (2 (\frac{1}{2} \cdot \sin (x)))$

Combine $\frac{1}{2}$ and $\sin (x)$ .

$g (\frac{1}{2} \cdot \sin (x)) = \arcsin (2 (\frac{\sin (x)}{2}))$

Cancel the common factor of $2$ .

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Cancel the common factor.

$g (\frac{1}{2} \cdot \sin (x)) = \arcsin (2 (\frac{\sin (x)}{2}))$

Rewrite the expression.

$g (\frac{1}{2} \cdot \sin (x)) = \arcsin (\sin (x))$

Since $g (f (x)) = x$ , $f^{- 1} (x) = \arcsin (2 x)$ is the inverse of $f (x) = \frac{1}{2} \cdot \sin (x)$ .

$f^{- 1} (x) = \arcsin (2 x)$

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Name

Name	one billion four hundred sixteen million three hundred seventy-six thousand six hundred fourteen

Interesting facts

1416376614 has 8 divisors, whose sum is 2832753240
The reverse of 1416376614 is 4166736141
Previous prime number is 3

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 39
Digital Root 3

Name

Name	three hundred ninety million seventy-seven thousand three hundred twenty-four

Interesting facts

390077324 has 16 divisors, whose sum is 887319576
The reverse of 390077324 is 423770093
Previous prime number is 91

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 35
Digital Root 8

Name

Name	nine hundred seventeen million eight hundred twenty-six thousand three hundred fifty

Interesting facts

917826350 has 64 divisors, whose sum is 1968645120
The reverse of 917826350 is 053628719
Previous prime number is 7

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 41
Digital Root 5

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