Trigonometry Examples

$\frac{1}{\sqrt{2} + \sec (x)}$

Step 1

Set the denominator in $\frac{1}{\sqrt{2} + \sec (x)}$ equal to $0$ to find where the expression is undefined.

$\sqrt{2} + \sec (x) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $\sqrt{2}$ from both sides of the equation.

$\sec (x) = - \sqrt{2}$

Step 2.2

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

$x = arcsec (- \sqrt{2})$

Step 2.3

Simplify the right side.

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Step 2.3.1

The exact value of $arcsec (- \sqrt{2})$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 2.4

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{3 π}{4}$

Step 2.5

Simplify $2 π - \frac{3 π}{4}$ .

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Step 2.5.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{3 π}{4}$

Step 2.5.2

Combine fractions.

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Step 2.5.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{3 π}{4}$

Step 2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Step 2.5.3

Simplify the numerator.

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Step 2.5.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - 3 π}{4}$

Step 2.5.3.2

Subtract $3 π$ from $8 π$ .

$x = \frac{5 π}{4}$

Step 2.6

Find the period of $\sec (x)$ .

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Step 2.6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 2.6.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.7

The period of the $\sec (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Step 3

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

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

Step 4

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

Set-Builder Notation:

${x | x \neq \frac{3 π}{4} + 2 π n, x \neq \frac{5 π}{4} + 2 π n, x \neq \frac{π}{2} + π n}$ , for any integer $n$

Step 5