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 one hundred eleven million two hundred twenty-six thousand nine hundred ninety-two

Interesting facts

1111226992 has 64 divisors, whose sum is 5807059776
The reverse of 1111226992 is 2996221111
Previous prime number is 31

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 34
Digital Root 7

Name

Name	two hundred ten million seven hundred fifty-four thousand one hundred seventy-five

Interesting facts

210754175 has 32 divisors, whose sum is 270729216
The reverse of 210754175 is 571457012
Previous prime number is 101

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 32
Digital Root 5

Name

Name	one billion one hundred fourteen million eight hundred twenty-two thousand four hundred seventy-seven

Interesting facts

1114822477 has 8 divisors, whose sum is 1121753600
The reverse of 1114822477 is 7742284111
Previous prime number is 5653

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 37
Digital Root 1

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