Trigonometry Examples

$\tan (\arccos (8 x))$

Step 1

Set the argument in $\arccos (8 x)$ greater than or equal to $- 1$ to find where the expression is defined.

$8 x \geq - 1$

Step 2

Divide each term in $8 x \geq - 1$ by $8$ and simplify.

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Step 2.1

Divide each term in $8 x \geq - 1$ by $8$ .

$\frac{8 x}{8} \geq \frac{- 1}{8}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $8$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} \geq \frac{- 1}{8}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 1}{8}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{1}{8}$

Step 3

Set the argument in $\arccos (8 x)$ less than or equal to $1$ to find where the expression is defined.

$8 x \leq 1$

Step 4

Divide each term in $8 x \leq 1$ by $8$ and simplify.

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Step 4.1

Divide each term in $8 x \leq 1$ by $8$ .

$\frac{8 x}{8} \leq \frac{1}{8}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $8$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{8 x}{8} \leq \frac{1}{8}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{1}{8}$

Step 5

Set the argument in $\tan (\arccos (8 x))$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$\arccos (8 x) = \frac{π}{2} + π n$ , for any integer $n$

Step 6

Solve for $x$ .

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Step 6.1

Take the inverse arccosine of both sides of the equation to extract $x$ from inside the arccosine.

$8 x = \cos (\frac{π}{2} + π n)$

Step 6.2

Divide each term in $8 x = \cos (\frac{π}{2} + π n)$ by $8$ and simplify.

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Step 6.2.1

Divide each term in $8 x = \cos (\frac{π}{2} + π n)$ by $8$ .

$\frac{8 x}{8} = \frac{\cos (\frac{π}{2} + π n)}{8}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $8$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{\cos (\frac{π}{2} + π n)}{8}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\cos (\frac{π}{2} + π n)}{8}$

Step 7

The domain is all values of $x$ that make the expression defined.

Set-Builder Notation:

${x | - \frac{1}{8} \leq x \leq \frac{1}{8}, x \neq \frac{\cos (\frac{π}{2} + π n)}{8}}$

Step 8