Find the Domain tan(arccos(8x))

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Trigonometry

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

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

$8 x \geq - 1$

Solve for $x$ .

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Divide each term by $8$ and simplify.

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

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

Cancel the common factor of $8$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Move the negative in front of the fraction.

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

The solution consists of all of the true intervals.

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

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

$8 x \leq 1$

Solve for $x$ .

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Divide each term by $8$ and simplify.

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

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

Cancel the common factor of $8$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

The solution consists of all of the true intervals.

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

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 $n$

Solve for $x$ .

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

$8 x = \cos (\frac{π}{2} + π n)$

Divide each term by $8$ and simplify.

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

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

Cancel the common factor of $8$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

Set-Builder Notation:

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

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Name

Name	one billion five hundred forty-six million five hundred sixty-five thousand nine

Interesting facts

1546565009 has 16 divisors, whose sum is 1850204160
The reverse of 1546565009 is 9005656451
Previous prime number is 11

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 41
Digital Root 5

Name

Name	eight hundred twenty million nine hundred fifty-one thousand six hundred two

Interesting facts

820951602 has 8 divisors, whose sum is 1641903216
The reverse of 820951602 is 206159028
Previous prime number is 3

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 33
Digital Root 6

Name

Name	one billion eight hundred eighty-two million two hundred thirty-nine thousand eight hundred thirty-eight

Interesting facts

1882239838 has 16 divisors, whose sum is 2949644160
The reverse of 1882239838 is 8389322881
Previous prime number is 101

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 52
Digital Root 7

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