Trigonometry Examples

$f (t) = - \tan (0.4 t)$

Step 1

Set the argument in $\tan (0.4 t)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$0.4 t = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Divide each term in $0.4 t = \frac{π}{2} + π n$ by $0.4$ and simplify.

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Step 2.1

Divide each term in $0.4 t = \frac{π}{2} + π n$ by $0.4$ .

$\frac{0.4 t}{0.4} = \frac{\frac{π}{2}}{0.4} + \frac{π n}{0.4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $0.4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{0.4 t}{0.4} = \frac{\frac{π}{2}}{0.4} + \frac{π n}{0.4}$

Step 2.2.1.2

Divide $t$ by $1$ .

$t = \frac{\frac{π}{2}}{0.4} + \frac{π n}{0.4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$t = \frac{π}{2} \cdot \frac{1}{0.4} + \frac{π n}{0.4}$

Step 2.3.1.2

Divide $1$ by $0.4$ .

$t = \frac{π}{2} \cdot 2.5 + \frac{π n}{0.4}$

Step 2.3.1.3

Multiply $\frac{π}{2} \cdot 2.5$ .

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Step 2.3.1.3.1

Combine $\frac{π}{2}$ and $2.5$ .

$t = \frac{π \cdot 2.5}{2} + \frac{π n}{0.4}$

Step 2.3.1.3.2

Multiply $π$ by $2.5$ .

$t = \frac{7.85398163}{2} + \frac{π n}{0.4}$

Step 2.3.1.4

Divide $7.85398163$ by $2$ .

$t = 3.92699081 + \frac{π n}{0.4}$

Step 2.3.1.5

Factor $π$ out of $π n$ .

$t = 3.92699081 + \frac{π (n)}{0.4}$

Step 2.3.1.6

Factor $0.4$ out of $0.4$ .

$t = 3.92699081 + \frac{π (n)}{0.4 (1)}$

Step 2.3.1.7

Separate fractions.

$t = 3.92699081 + \frac{π}{0.4} \cdot \frac{n}{1}$

Step 2.3.1.8

Replace $π$ with an approximation.

$t = 3.92699081 + \frac{3.14159265}{0.4} \cdot \frac{n}{1}$

Step 2.3.1.9

Divide $3.14159265$ by $0.4$ .

$t = 3.92699081 + 7.85398163 \frac{n}{1}$

Step 2.3.1.10

Divide $n$ by $1$ .

$t = 3.92699081 + 7.85398163 n$

Step 3

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

Set-Builder Notation:

${t | t \neq 3.92699081 + 7.85398163 n}$ , for any integer $n$

Step 4