Trigonometry Examples

$f (t) = \frac{1}{2} \cdot \sec (t + 2) - π$

Step 1

Write $f (t) = \frac{1}{2} \cdot \sec (t + 2) - π$ as an equation.

$y = \frac{1}{2} \cdot \sec (t + 2) - π$

Step 2

Interchange the variables.

$t = \frac{1}{2} \cdot \sec (y + 2) - π$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{1}{2} \cdot \sec (y + 2) - π = t$ .

$\frac{1}{2} \cdot \sec (y + 2) - π = t$

Step 3.2

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

$\frac{\sec (y + 2)}{2} - π = t$

Step 3.3

Add $π$ to both sides of the equation.

$\frac{\sec (y + 2)}{2} = t + π$

Step 3.4

Multiply both sides of the equation by $2$ .

$2 \frac{\sec (y + 2)}{2} = 2 (t + π)$

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Cancel the common factor of $2$ .

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Step 3.5.1.1.1

Cancel the common factor.

$2 \frac{\sec (y + 2)}{2} = 2 (t + π)$

Step 3.5.1.1.2

Rewrite the expression.

$\sec (y + 2) = 2 (t + π)$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Apply the distributive property.

$\sec (y + 2) = 2 t + 2 π$

Step 3.6

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

$y + 2 = arcsec (2 t + 2 π)$

Step 3.7

Subtract $2$ from both sides of the equation.

$y = arcsec (2 t + 2 π) - 2$

Step 4

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

$f^{- 1} (t) = arcsec (2 t + 2 π) - 2$

Step 5

Verify if $f^{- 1} (t) = arcsec (2 t + 2 π) - 2$ is the inverse of $f (t) = \frac{1}{2} \cdot \sec (t + 2) - π$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{1}{2} \cdot \sec (t + 2) - π)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{1}{2} \cdot \sec (t + 2) - π) = arcsec (2 (\frac{1}{2} \cdot \sec (t + 2) - π) + 2 π) - 2$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Combine $\frac{1}{2}$ and $\sec (t + 2)$ .

$f^{- 1} (\frac{1}{2} \cdot \sec (t + 2) - π) = arcsec (2 (\frac{\sec (t + 2)}{2} - π) + 2 π) - 2$

Step 5.2.3.1.2

Apply the distributive property.

$f^{- 1} (\frac{1}{2} \cdot \sec (t + 2) - π) = arcsec (2 (\frac{\sec (t + 2)}{2}) + 2 (- π) + 2 π) - 2$

Step 5.2.3.1.3

Cancel the common factor of $2$ .

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Step 5.2.3.1.3.1

Cancel the common factor.

$f^{- 1} (\frac{1}{2} \cdot \sec (t + 2) - π) = arcsec (2 (\frac{\sec (t + 2)}{2}) + 2 (- π) + 2 π) - 2$

Step 5.2.3.1.3.2

Rewrite the expression.

$f^{- 1} (\frac{1}{2} \cdot \sec (t + 2) - π) = arcsec (\sec (t + 2) + 2 (- π) + 2 π) - 2$

Step 5.2.3.1.4

Multiply $- 1$ by $2$ .

$f^{- 1} (\frac{1}{2} \cdot \sec (t + 2) - π) = arcsec (\sec (t + 2) - 2 π + 2 π) - 2$

Step 5.2.3.2

Combine the opposite terms in $\sec (t + 2) - 2 π + 2 π$ .

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Step 5.2.3.2.1

Add $- 2 π$ and $2 π$ .

$f^{- 1} (\frac{1}{2} \cdot \sec (t + 2) - π) = arcsec (\sec (t + 2) + 0) - 2$

Step 5.2.3.2.2

Add $\sec (t + 2)$ and $0$ .

$f^{- 1} (\frac{1}{2} \cdot \sec (t + 2) - π) = arcsec (\sec (t + 2)) - 2$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (arcsec (2 t + 2 π) - 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (arcsec (2 t + 2 π) - 2) = \frac{1}{2} \cdot \sec ((arcsec (2 t + 2 π) - 2) + 2) - π$

Step 5.3.3

Combine the opposite terms in $\frac{1}{2} \cdot \sec ((arcsec (2 t + 2 π) - 2) + 2) - π$ .

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Step 5.3.3.1

Add $- 2$ and $2$ .

$f (arcsec (2 t + 2 π) - 2) = \frac{1}{2} \cdot \sec (arcsec (2 t + 2 π) + 0) - π$

Step 5.3.3.2

Add $arcsec (2 t + 2 π)$ and $0$ .

$f (arcsec (2 t + 2 π) - 2) = \frac{1}{2} \cdot \sec (arcsec (2 t + 2 π)) - π$

Step 5.3.4

Simplify each term.

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Step 5.3.4.1

The functions secant and arcsecant are inverses.

$f (arcsec (2 t + 2 π) - 2) = \frac{1}{2} \cdot (2 t + 2 π) - π$

Step 5.3.4.2

Apply the distributive property.

$f (arcsec (2 t + 2 π) - 2) = \frac{1}{2} \cdot (2 t) + \frac{1}{2} \cdot (2 π) - π$

Step 5.3.4.3

Cancel the common factor of $2$ .

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Step 5.3.4.3.1

Factor $2$ out of $2 t$ .

$f (arcsec (2 t + 2 π) - 2) = \frac{1}{2} \cdot (2 (t)) + \frac{1}{2} \cdot (2 π) - π$

Step 5.3.4.3.2

Cancel the common factor.

$f (arcsec (2 t + 2 π) - 2) = \frac{1}{2} \cdot (2 t) + \frac{1}{2} \cdot (2 π) - π$

Step 5.3.4.3.3

Rewrite the expression.

$f (arcsec (2 t + 2 π) - 2) = t + \frac{1}{2} \cdot (2 π) - π$

Step 5.3.4.4

Cancel the common factor of $2$ .

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Step 5.3.4.4.1

Factor $2$ out of $2 π$ .

$f (arcsec (2 t + 2 π) - 2) = t + \frac{1}{2} \cdot (2 (π)) - π$

Step 5.3.4.4.2

Cancel the common factor.

$f (arcsec (2 t + 2 π) - 2) = t + \frac{1}{2} \cdot (2 π) - π$

Step 5.3.4.4.3

Rewrite the expression.

$f (arcsec (2 t + 2 π) - 2) = t + π - π$

Step 5.3.5

Combine the opposite terms in $t + π - π$ .

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Step 5.3.5.1

Subtract $π$ from $π$ .

$f (arcsec (2 t + 2 π) - 2) = t + 0$

Step 5.3.5.2

Add $t$ and $0$ .

$f (arcsec (2 t + 2 π) - 2) = t$

Step 5.4

Since $f^{- 1} (f (t)) = t$ and $f (f^{- 1} (t)) = t$ , then $f^{- 1} (t) = arcsec (2 t + 2 π) - 2$ is the inverse of $f (t) = \frac{1}{2} \cdot \sec (t + 2) - π$ .

$f^{- 1} (t) = arcsec (2 t + 2 π) - 2$