Trigonometry Examples

$\tan (\arccos (10 x))$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the argument in $\arccos (10 x)$ greater than or equal to $- 1$ to find where the expression is defined.

$10 x \geq - 1$

Step 1.2

Divide each term in $10 x \geq - 1$ by $10$ and simplify.

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Step 1.2.1

Divide each term in $10 x \geq - 1$ by $10$ .

$\frac{10 x}{10} \geq \frac{- 1}{10}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $10$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} \geq \frac{- 1}{10}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 1}{10}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{1}{10}$

Step 1.3

Set the argument in $\arccos (10 x)$ less than or equal to $1$ to find where the expression is defined.

$10 x \leq 1$

Step 1.4

Divide each term in $10 x \leq 1$ by $10$ and simplify.

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Step 1.4.1

Divide each term in $10 x \leq 1$ by $10$ .

$\frac{10 x}{10} \leq \frac{1}{10}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of $10$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{10 x}{10} \leq \frac{1}{10}$

Step 1.4.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{1}{10}$

Step 1.5

Set the argument in $\tan (\arccos (10 x))$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$\arccos (10 x) = \frac{π}{2} + π n$ , for any integer $n$

Step 1.6

Solve for $x$ .

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Step 1.6.1

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

$10 x = \cos (\frac{π}{2} + π n)$

Step 1.6.2

Divide each term in $10 x = \cos (\frac{π}{2} + π n)$ by $10$ and simplify.

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Step 1.6.2.1

Divide each term in $10 x = \cos (\frac{π}{2} + π n)$ by $10$ .

$\frac{10 x}{10} = \frac{\cos (\frac{π}{2} + π n)}{10}$

Step 1.6.2.2

Simplify the left side.

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Step 1.6.2.2.1

Cancel the common factor of $10$ .

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Step 1.6.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{\cos (\frac{π}{2} + π n)}{10}$

Step 1.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\cos (\frac{π}{2} + π n)}{10}$

Step 1.7

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

Set-Builder Notation:

${x | - \frac{1}{10} \leq x \leq \frac{1}{10}, x \neq \frac{\cos (\frac{π}{2} + π n)}{10}}$

Set-Builder Notation:

${x | - \frac{1}{10} \leq x \leq \frac{1}{10}, x \neq \frac{\cos (\frac{π}{2} + π n)}{10}}$

Step 2

Since the domain is not all real numbers, $\tan (\arccos (10 x))$ is not continuous over all real numbers.

Not continuous

Step 3