Trigonometry Examples

$y = \tan (- \frac{3}{4} x)$

Step 1

Interchange the variables.

$x = \tan (- \frac{3}{4} y)$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $\tan (- \frac{3}{4} y) = x$ .

$\tan (- \frac{3}{4} y) = x$

Step 2.2

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

$- \frac{3}{4} y = \arctan (x)$

Step 2.3

Simplify the left side.

Tap for more steps...

Step 2.3.1

Simplify $- \frac{3}{4} y$ .

Tap for more steps...

Step 2.3.1.1

Combine $y$ and $\frac{3}{4}$ .

$- \frac{y \cdot 3}{4} = \arctan (x)$

Step 2.3.1.2

Move $3$ to the left of $y$ .

$- \frac{3 y}{4} = \arctan (x)$

Step 2.4

Multiply both sides of the equation by $- \frac{4}{3}$ .

$- \frac{4}{3} (- \frac{3 y}{4}) = - \frac{4}{3} \arctan (x)$

Step 2.5

Simplify both sides of the equation.

Tap for more steps...

Step 2.5.1

Simplify the left side.

Tap for more steps...

Step 2.5.1.1

Simplify $- \frac{4}{3} (- \frac{3 y}{4})$ .

Tap for more steps...

Step 2.5.1.1.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.5.1.1.1.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$\frac{- 4}{3} (- \frac{3 y}{4}) = - \frac{4}{3} \arctan (x)$

Step 2.5.1.1.1.2

Move the leading negative in $- \frac{3 y}{4}$ into the numerator.

$\frac{- 4}{3} \cdot \frac{- 3 y}{4} = - \frac{4}{3} \arctan (x)$

Step 2.5.1.1.1.3

Factor $4$ out of $- 4$ .

$\frac{4 (- 1)}{3} \cdot \frac{- 3 y}{4} = - \frac{4}{3} \arctan (x)$

Step 2.5.1.1.1.4

Cancel the common factor.

$\frac{4 \cdot - 1}{3} \cdot \frac{- 3 y}{4} = - \frac{4}{3} \arctan (x)$

Step 2.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 y) = - \frac{4}{3} \arctan (x)$

Step 2.5.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.5.1.1.2.1

Factor $3$ out of $- 3 y$ .

$\frac{- 1}{3} (3 (- y)) = - \frac{4}{3} \arctan (x)$

Step 2.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- y)) = - \frac{4}{3} \arctan (x)$

Step 2.5.1.1.2.3

Rewrite the expression.

$- - y = - \frac{4}{3} \arctan (x)$

Step 2.5.1.1.3

Multiply.

Tap for more steps...

Step 2.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{4}{3} \arctan (x)$

Step 2.5.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{4}{3} \arctan (x)$

Step 2.5.2

Simplify the right side.

Tap for more steps...

Step 2.5.2.1

Simplify $- \frac{4}{3} \arctan (x)$ .

Tap for more steps...

Step 2.5.2.1.1

Combine $\arctan (x)$ and $\frac{4}{3}$ .

$y = - \frac{\arctan (x) \cdot 4}{3}$

Step 2.5.2.1.2

Move $4$ to the left of $\arctan (x)$ .

$y = - \frac{4 \arctan (x)}{3}$

Step 3

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

$f^{- 1} (x) = - \frac{4 \arctan (x)}{3}$

Step 4

Verify if $f^{- 1} (x) = - \frac{4 \arctan (x)}{3}$ is the inverse of $f (x) = \tan (- \frac{3}{4} x)$ .

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 $f^{- 1} (\tan (- \frac{3}{4} x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\tan (- \frac{3}{4} x)) = - \frac{4 \arctan (\tan (- \frac{3}{4} x))}{3}$

Step 4.2.3

Simplify the numerator.

Tap for more steps...

Step 4.2.3.1

Combine $x$ and $\frac{3}{4}$ .

$f^{- 1} (\tan (- \frac{3}{4} x)) = - \frac{4 \arctan (\tan (- \frac{x \cdot 3}{4}))}{3}$

Step 4.2.3.2

Move $3$ to the left of $x$ .

$f^{- 1} (\tan (- \frac{3}{4} x)) = - \frac{4 \arctan (\tan (- \frac{3 \cdot x}{4}))}{3}$

Step 4.2.3.3

Since $\tan (- \frac{3 x}{4})$ is an odd function, rewrite $\tan (- \frac{3 x}{4})$ as $- \tan (\frac{3 x}{4})$ .

$f^{- 1} (\tan (- \frac{3}{4} x)) = - \frac{4 \arctan (- \tan (\frac{3 x}{4}))}{3}$

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 $f (- \frac{4 \arctan (x)}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{4 \arctan (x)}{3}) = \tan (- \frac{3}{4} \cdot (- \frac{4 \arctan (x)}{3}))$

Step 4.3.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.3.3.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$f (- \frac{4 \arctan (x)}{3}) = \tan (\frac{- 3}{4} \cdot (- \frac{4 \arctan (x)}{3}))$

Step 4.3.3.2

Move the leading negative in $- \frac{4 \arctan (x)}{3}$ into the numerator.

$f (- \frac{4 \arctan (x)}{3}) = \tan (\frac{- 3}{4} \cdot \frac{- 4 \arctan (x)}{3})$

Step 4.3.3.3

Factor $3$ out of $- 3$ .

$f (- \frac{4 \arctan (x)}{3}) = \tan (\frac{3 (- 1)}{4} \cdot \frac{- 4 \arctan (x)}{3})$

Step 4.3.3.4

Cancel the common factor.

$f (- \frac{4 \arctan (x)}{3}) = \tan (\frac{3 \cdot - 1}{4} \cdot \frac{- 4 \arctan (x)}{3})$

Step 4.3.3.5

Rewrite the expression.

$f (- \frac{4 \arctan (x)}{3}) = \tan (\frac{- 1}{4} \cdot (- 4 \arctan (x)))$

Step 4.3.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.3.4.1

Factor $4$ out of $- 4 \arctan (x)$ .

$f (- \frac{4 \arctan (x)}{3}) = \tan (\frac{- 1}{4} \cdot (4 (- \arctan (x))))$

Step 4.3.4.2

Cancel the common factor.

$f (- \frac{4 \arctan (x)}{3}) = \tan (\frac{- 1}{4} \cdot (4 (- \arctan (x))))$

Step 4.3.4.3

Rewrite the expression.

$f (- \frac{4 \arctan (x)}{3}) = \tan (- 1 \cdot (- \arctan (x)))$

Step 4.3.5

Multiply.

Tap for more steps...

Step 4.3.5.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{4 \arctan (x)}{3}) = \tan (1 \cdot \arctan (x))$

Step 4.3.5.2

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

$f (- \frac{4 \arctan (x)}{3}) = \tan (\arctan (x))$

Step 4.3.6

The functions tangent and arctangent are inverses.

$f (- \frac{4 \arctan (x)}{3}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{4 \arctan (x)}{3}$ is the inverse of $f (x) = \tan (- \frac{3}{4} x)$ .

$f^{- 1} (x) = - \frac{4 \arctan (x)}{3}$