Trigonometry Examples

$\tan (\arcsin (x))$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

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

$x \geq - 1$

Step 1.2

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

$x \leq 1$

Step 1.3

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

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

Step 1.4

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

$x = \sin (\frac{π}{2} + π n)$

Step 1.5

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

Set-Builder Notation:

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

Set-Builder Notation:

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

Step 2

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

Not continuous

Step 3