Trigonometry Examples

$\frac{2 \sin (y)}{\sin (2 y)} + \frac{1}{\cos (y)}$

Step 1

Rewrite the equation as $\frac{2 \sin (y)}{\sin (2 y)} + \frac{1}{\cos (y)} = x$ .

$\frac{2 \sin (y)}{\sin (2 y)} + \frac{1}{\cos (y)} = x$

Step 2

Simplify the left side.

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Step 2.1

Simplify $\frac{2 \sin (y)}{\sin (2 y)} + \frac{1}{\cos (y)}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the sine double-angle identity.

$\frac{2 \sin (y)}{2 \sin (y) \cos (y)} + \frac{1}{\cos (y)} = x$

Step 2.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.2.1

Cancel the common factor.

$\frac{2 \sin (y)}{2 \sin (y) \cos (y)} + \frac{1}{\cos (y)} = x$

Step 2.1.1.2.2

Rewrite the expression.

$\frac{\sin (y)}{\sin (y) \cos (y)} + \frac{1}{\cos (y)} = x$

Step 2.1.1.3

Cancel the common factor of $\sin (y)$ .

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Step 2.1.1.3.1

Cancel the common factor.

$\frac{\sin (y)}{\sin (y) \cos (y)} + \frac{1}{\cos (y)} = x$

Step 2.1.1.3.2

Rewrite the expression.

$\frac{1}{\cos (y)} + \frac{1}{\cos (y)} = x$

Step 2.1.1.4

Convert from $\frac{1}{\cos (y)}$ to $\sec (y)$ .

$\sec (y) + \frac{1}{\cos (y)} = x$

Step 2.1.1.5

Convert from $\frac{1}{\cos (y)}$ to $\sec (y)$ .

$\sec (y) + \sec (y) = x$

Step 2.1.2

Add $\sec (y)$ and $\sec (y)$ .

$2 \sec (y) = x$

Step 3

Divide each term in $2 \sec (y) = x$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 \sec (y) = x$ by $2$ .

$\frac{2 \sec (y)}{2} = \frac{x}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 \sec (y)}{2} = \frac{x}{2}$

Step 3.2.1.2

Divide $\sec (y)$ by $1$ .

$\sec (y) = \frac{x}{2}$

Step 4

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

$y = arcsec (\frac{x}{2})$

Step 5

Interchange the variables.

$x = arcsec (\frac{y}{2})$

Step 6

Solve for $y$ .

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Step 6.1

Rewrite the equation as $arcsec (\frac{y}{2}) = x$ .

$arcsec (\frac{y}{2}) = x$

Step 6.2

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

$\frac{y}{2} = \sec (x)$

Step 6.3

Multiply both sides of the equation by $2$ .

$2 \frac{y}{2} = 2 \sec (x)$

Step 6.4

Simplify the left side.

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Step 6.4.1

Cancel the common factor of $2$ .

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Step 6.4.1.1

Cancel the common factor.

$2 \frac{y}{2} = 2 \sec (x)$

Step 6.4.1.2

Rewrite the expression.

$y = 2 \sec (x)$

Step 7

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

$f^{- 1} (x) = 2 \sec (x)$

Step 8

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

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Step 8.1

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

Step 8.2

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

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Step 8.2.1

Set up the composite result function.

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

Step 8.2.2

Evaluate $f^{- 1} (arcsec (\frac{x}{2}))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (arcsec (\frac{x}{2})) = 2 \sec (arcsec (\frac{x}{2}))$

Step 8.2.3

The functions secant and arcsecant are inverses.

$f^{- 1} (arcsec (\frac{x}{2})) = 2 (\frac{x}{2})$

Step 8.2.4

Cancel the common factor of $2$ .

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Step 8.2.4.1

Cancel the common factor.

$f^{- 1} (arcsec (\frac{x}{2})) = 2 (\frac{x}{2})$

Step 8.2.4.2

Rewrite the expression.

$f^{- 1} (arcsec (\frac{x}{2})) = x$

Step 8.3

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

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Step 8.3.1

Set up the composite result function.

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

Step 8.3.2

Evaluate $f (2 \sec (x))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (2 \sec (x)) = arcsec (\frac{2 \sec (x)}{2})$

Step 8.3.3

Cancel the common factor of $2$ .

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Step 8.3.3.1

Cancel the common factor.

$f (2 \sec (x)) = arcsec (\frac{2 \sec (x)}{2})$

Step 8.3.3.2

Divide $\sec (x)$ by $1$ .

$f (2 \sec (x)) = arcsec (\sec (x))$

Step 8.4

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

$f^{- 1} (x) = 2 \sec (x)$