Finite Math Examples

$y = \sin (e^{- x})$

Step 1

Rewrite the equation as $\sin (e^{- x}) = y$ .

$\sin (e^{- x}) = y$

Step 2

Substitute $u$ for $e^{- x}$ .

$\sin (u) = y$

Step 3

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

$u = \arcsin (y)$

Step 4

Substitute $e^{- x}$ for $u$ and solve $e^{- x} = \arcsin (y)$

Tap for more steps...

Step 4.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- x}) = \ln (\arcsin (y))$

Step 4.2

Expand the left side.

Tap for more steps...

Step 4.2.1

Expand $\ln (e^{- x})$ by moving $- x$ outside the logarithm.

$- x \ln (e) = \ln (\arcsin (y))$

Step 4.2.2

The natural logarithm of $e$ is $1$ .

$- x \cdot 1 = \ln (\arcsin (y))$

Step 4.2.3

Multiply $- 1$ by $1$ .

$- x = \ln (\arcsin (y))$

Step 4.3

Divide each term in $- x = \ln (\arcsin (y))$ by $- 1$ and simplify.

Tap for more steps...

Step 4.3.1

Divide each term in $- x = \ln (\arcsin (y))$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\ln (\arcsin (y))}{- 1}$

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\ln (\arcsin (y))}{- 1}$

Step 4.3.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (\arcsin (y))}{- 1}$

Step 4.3.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1

Move the negative one from the denominator of $\frac{\ln (\arcsin (y))}{- 1}$ .

$x = - 1 \cdot \ln (\arcsin (y))$

Step 4.3.3.2

Rewrite $- 1 \cdot \ln (\arcsin (y))$ as $- \ln (\arcsin (y))$ .

$x = - \ln (\arcsin (y))$