Trigonometry Examples

y = \sin (x)

$y = \sin (x)$

Step 1

Interchange the variables.

x = \sin (y)

$x = \sin (y)$

Step 2

Solve for

y

$y$ .

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Step 2.1

Rewrite the equation as

\sin (y) = x

$\sin (y) = x$ .

\sin (y) = x

$\sin (y) = x$

Step 2.2

Take the inverse sine of both sides of the equation to extract

y

$y$ from inside the sine.

y = \arcsin (x)

$y = \arcsin (x)$

Step 2.3

Remove parentheses.

y = \arcsin (x)

$y = \arcsin (x)$

y = \arcsin (x)

$y = \arcsin (x)$

Step 3

Replace

y

$y$ with

f^{- 1} (x)

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

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

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

Step 4

Verify if

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

$f^{- 1} (x) = \arcsin (x)$ is the inverse of

f (x) = \sin (x)

$f (x) = \sin (x)$ .

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Step 4.1

To verify the inverse, check if

f^{- 1} (f (x)) = x

$f^{- 1} (f (x)) = x$ and

f (f^{- 1} (x)) = x

$f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate

f^{- 1} (f (x))

$f^{- 1} (f (x))$ .

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Step 4.2.1

Set up the composite result function.

f^{- 1} (f (x))

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate

f^{- 1} (\sin (x))

$f^{- 1} (\sin (x))$ by substituting in the value of

f

$f$ into

f^{- 1}

$f^{- 1}$ .

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

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

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

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

Step 4.3

Evaluate

f (f^{- 1} (x))

$f (f^{- 1} (x))$ .

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Step 4.3.1

Set up the composite result function.

f (f^{- 1} (x))

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate

f (\arcsin (x))

$f (\arcsin (x))$ by substituting in the value of

f^{- 1}

$f^{- 1}$ into

f

$f$ .

f (\arcsin (x)) = \sin (\arcsin (x))

$f (\arcsin (x)) = \sin (\arcsin (x))$

Step 4.3.3

The functions sine and arcsine are inverses.

f (\arcsin (x)) = x

$f (\arcsin (x)) = x$

f (\arcsin (x)) = x

$f (\arcsin (x)) = x$

Step 4.4

Since

f^{- 1} (f (x)) = x

$f^{- 1} (f (x)) = x$ and

f (f^{- 1} (x)) = x

$f (f^{- 1} (x)) = x$ , then

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

$f^{- 1} (x) = \arcsin (x)$ is the inverse of

f (x) = \sin (x)

$f (x) = \sin (x)$ .

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

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

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

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

[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$