Trigonometry Examples

$\tan (- \frac{5 x}{6})$

Step 1

Interchange the variables.

$x = \tan (- \frac{5 y}{6})$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $\tan (- \frac{5 y}{6}) = x$ .

$\tan (- \frac{5 y}{6}) = x$

Step 2.2

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

$- \frac{5 y}{6} = \arctan (x)$

Step 2.3

Multiply both sides of the equation by $- \frac{6}{5}$ .

$- \frac{6}{5} (- \frac{5 y}{6}) = - \frac{6}{5} \arctan (x)$

Step 2.4

Simplify both sides of the equation.

Tap for more steps...

Step 2.4.1

Simplify the left side.

Tap for more steps...

Step 2.4.1.1

Simplify $- \frac{6}{5} (- \frac{5 y}{6})$ .

Tap for more steps...

Step 2.4.1.1.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 2.4.1.1.1.1

Move the leading negative in $- \frac{6}{5}$ into the numerator.

$\frac{- 6}{5} (- \frac{5 y}{6}) = - \frac{6}{5} \arctan (x)$

Step 2.4.1.1.1.2

Move the leading negative in $- \frac{5 y}{6}$ into the numerator.

$\frac{- 6}{5} \cdot \frac{- 5 y}{6} = - \frac{6}{5} \arctan (x)$

Step 2.4.1.1.1.3

Factor $6$ out of $- 6$ .

$\frac{6 (- 1)}{5} \cdot \frac{- 5 y}{6} = - \frac{6}{5} \arctan (x)$

Step 2.4.1.1.1.4

Cancel the common factor.

$\frac{6 \cdot - 1}{5} \cdot \frac{- 5 y}{6} = - \frac{6}{5} \arctan (x)$

Step 2.4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{5} (- 5 y) = - \frac{6}{5} \arctan (x)$

Step 2.4.1.1.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 2.4.1.1.2.1

Factor $5$ out of $- 5 y$ .

$\frac{- 1}{5} (5 (- y)) = - \frac{6}{5} \arctan (x)$

Step 2.4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{5} (5 (- y)) = - \frac{6}{5} \arctan (x)$

Step 2.4.1.1.2.3

Rewrite the expression.

$- - y = - \frac{6}{5} \arctan (x)$

Step 2.4.1.1.3

Multiply.

Tap for more steps...

Step 2.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{6}{5} \arctan (x)$

Step 2.4.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{6}{5} \arctan (x)$

Step 2.4.2

Simplify the right side.

Tap for more steps...

Step 2.4.2.1

Simplify $- \frac{6}{5} \arctan (x)$ .

Tap for more steps...

Step 2.4.2.1.1

Combine $\arctan (x)$ and $\frac{6}{5}$ .

$y = - \frac{\arctan (x) \cdot 6}{5}$

Step 2.4.2.1.2

Move $6$ to the left of $\arctan (x)$ .

$y = - \frac{6 \arctan (x)}{5}$

Step 3

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

$f^{- 1} (x) = - \frac{6 \arctan (x)}{5}$

Step 4

Verify if $f^{- 1} (x) = - \frac{6 \arctan (x)}{5}$ is the inverse of $f (x) = \tan (- \frac{5 x}{6})$ .

Tap for more steps...

Step 4.1

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\tan (- \frac{5 x}{6}))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\tan (- \frac{5 x}{6})) = - \frac{6 \arctan (\tan (- \frac{5 x}{6}))}{5}$

Step 4.2.3

Since $\tan (- \frac{5 x}{6})$ is an odd function, rewrite $\tan (- \frac{5 x}{6})$ as $- \tan (\frac{5 x}{6})$ .

$f^{- 1} (\tan (- \frac{5 x}{6})) = - \frac{6 \arctan (- \tan (\frac{5 x}{6}))}{5}$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (- \frac{6 \arctan (x)}{5})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{6 \arctan (x)}{5}) = \tan (- \frac{5 (- \frac{6 \arctan (x)}{5})}{6})$

Step 4.3.3

Simplify the numerator.

Tap for more steps...

Step 4.3.3.1

Multiply $- 1$ by $5$ .

$f (- \frac{6 \arctan (x)}{5}) = \tan (- \frac{- 5 \frac{6 \arctan (x)}{5}}{6})$

Step 4.3.3.2

Combine $- 5$ and $\frac{6 \arctan (x)}{5}$ .

$f (- \frac{6 \arctan (x)}{5}) = \tan (- \frac{\frac{- 5 (6 \arctan (x))}{5}}{6})$

Step 4.3.4

Multiply $- 5$ by $6$ .

$f (- \frac{6 \arctan (x)}{5}) = \tan (- \frac{\frac{- 30 \arctan (x)}{5}}{6})$

Step 4.3.5

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.3.5.1

Reduce the expression $\frac{- 30 \arctan (x)}{5}$ by cancelling the common factors.

Tap for more steps...

Step 4.3.5.1.1

Factor $5$ out of $- 30 \arctan (x)$ .

$f (- \frac{6 \arctan (x)}{5}) = \tan (- \frac{\frac{5 (- 6 \arctan (x))}{5}}{6})$

Step 4.3.5.1.2

Factor $5$ out of $5$ .

$f (- \frac{6 \arctan (x)}{5}) = \tan (- \frac{\frac{5 (- 6 \arctan (x))}{5 (1)}}{6})$

Step 4.3.5.1.3

Cancel the common factor.

$f (- \frac{6 \arctan (x)}{5}) = \tan (- \frac{\frac{5 (- 6 \arctan (x))}{5 \cdot 1}}{6})$

Step 4.3.5.1.4

Rewrite the expression.

$f (- \frac{6 \arctan (x)}{5}) = \tan (- \frac{\frac{- 6 \arctan (x)}{1}}{6})$

Step 4.3.5.2

Divide $- 6 \arctan (x)$ by $1$ .

$f (- \frac{6 \arctan (x)}{5}) = \tan (- \frac{- 6 \arctan (x)}{6})$

Step 4.3.6

Cancel the common factor of $- 6$ and $6$ .

Tap for more steps...

Step 4.3.6.1

Factor $6$ out of $- 6 \arctan (x)$ .

$f (- \frac{6 \arctan (x)}{5}) = \tan (- \frac{6 (- \arctan (x))}{6})$

Step 4.3.6.2

Cancel the common factors.

Tap for more steps...

Step 4.3.6.2.1

Factor $6$ out of $6$ .

$f (- \frac{6 \arctan (x)}{5}) = \tan (- \frac{6 (- \arctan (x))}{6 (1)})$

Step 4.3.6.2.2

Cancel the common factor.

$f (- \frac{6 \arctan (x)}{5}) = \tan (- \frac{6 (- \arctan (x))}{6 \cdot 1})$

Step 4.3.6.2.3

Rewrite the expression.

$f (- \frac{6 \arctan (x)}{5}) = \tan (- \frac{- \arctan (x)}{1})$

Step 4.3.6.2.4

Divide $- \arctan (x)$ by $1$ .

$f (- \frac{6 \arctan (x)}{5}) = \tan (\arctan (x))$

Step 4.3.7

The functions tangent and arctangent are inverses.

$f (- \frac{6 \arctan (x)}{5}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{6 \arctan (x)}{5}$ is the inverse of $f (x) = \tan (- \frac{5 x}{6})$ .

$f^{- 1} (x) = - \frac{6 \arctan (x)}{5}$