Trigonometry Examples

$\sin (180 - x)$

Step 1

Interchange the variables.

$x = \sin (180 - y)$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\sin (180 - y) = x$ .

$\sin (180 - y) = x$

Step 2.2

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

$180 - y = \arcsin (x)$

Step 2.3

Subtract $180$ from both sides of the equation.

$- y = \arcsin (x) - 180$

Step 2.4

Divide each term in $- y = \arcsin (x) - 180$ by $- 1$ and simplify.

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Step 2.4.1

Divide each term in $- y = \arcsin (x) - 180$ by $- 1$ .

$\frac{- y}{- 1} = \frac{\arcsin (x)}{- 1} + \frac{- 180}{- 1}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{\arcsin (x)}{- 1} + \frac{- 180}{- 1}$

Step 2.4.2.2

Divide $y$ by $1$ .

$y = \frac{\arcsin (x)}{- 1} + \frac{- 180}{- 1}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

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

$y = - 1 \cdot \arcsin (x) + \frac{- 180}{- 1}$

Step 2.4.3.1.2

Rewrite $- 1 \cdot \arcsin (x)$ as $- \arcsin (x)$ .

$y = - \arcsin (x) + \frac{- 180}{- 1}$

Step 2.4.3.1.3

Divide $- 180$ by $- 1$ .

$y = - \arcsin (x) + 180$

Step 3

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

$f^{- 1} (x) = - \arcsin (x) + 180$

Step 4

Verify if $f^{- 1} (x) = - \arcsin (x) + 180$ is the inverse of $f (x) = \sin (180 - x)$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\sin (180 - x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sin (180 - x)) = - \arcsin (\sin (180 - x)) + 180$

Step 4.2.3

Reorder $180$ and $- x$ .

$f^{- 1} (\sin (180 - x)) = - \arcsin (\sin (- x + 180)) + 180$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (- \arcsin (x) + 180)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \arcsin (x) + 180) = \sin (180 - (- \arcsin (x) + 180))$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$f (- \arcsin (x) + 180) = \sin (180 + \arcsin (x) - 1 \cdot 180)$

Step 4.3.3.2

Multiply $- - \arcsin (x)$ .

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Step 4.3.3.2.1

Multiply $- 1$ by $- 1$ .

$f (- \arcsin (x) + 180) = \sin (180 + 1 \arcsin (x) - 1 \cdot 180)$

Step 4.3.3.2.2

Multiply $\arcsin (x)$ by $1$ .

$f (- \arcsin (x) + 180) = \sin (180 + \arcsin (x) - 1 \cdot 180)$

Step 4.3.3.3

Multiply $- 1$ by $180$ .

$f (- \arcsin (x) + 180) = \sin (180 + \arcsin (x) - 180)$

Step 4.3.4

Combine the opposite terms in $180 + \arcsin (x) - 180$ .

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Step 4.3.4.1

Subtract $180$ from $180$ .

$f (- \arcsin (x) + 180) = \sin (0 + \arcsin (x))$

Step 4.3.4.2

Add $0$ and $\arcsin (x)$ .

$f (- \arcsin (x) + 180) = \sin (\arcsin (x))$

Step 4.3.5

The functions sine and arcsine are inverses.

$f (- \arcsin (x) + 180) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \arcsin (x) + 180$ is the inverse of $f (x) = \sin (180 - x)$ .

$f^{- 1} (x) = - \arcsin (x) + 180$