Trigonometry Examples

y = sec (2 x + \frac{π}{4})

$y = \sec (2 x + \frac{π}{4})$

Step 1

Set the argument in

sec (2 x + \frac{π}{4})

$\sec (2 x + \frac{π}{4})$ equal to

\frac{π}{2} + π n

$\frac{π}{2} + π n$ to find where the expression is undefined.

2 x + \frac{π}{4} = \frac{π}{2} + π n

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

n

$n$

Step 2

Solve for

x

$x$ .

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Step 2.1

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 2.1.1

Subtract

\frac{π}{4}

$\frac{π}{4}$ from both sides of the equation.

2 x = \frac{π}{2} + π n - \frac{π}{4}

$2 x = \frac{π}{2} + π n - \frac{π}{4}$

Step 2.1.2

To write

\frac{π}{2}

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

\frac{2}{2}

$\frac{2}{2}$ .

2 x = π n + \frac{π}{2} \cdot \frac{2}{2} - \frac{π}{4}

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

Step 2.1.3

Write each expression with a common denominator of

4

$4$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 2.1.3.1

Multiply

\frac{π}{2}

$\frac{π}{2}$ by

\frac{2}{2}

$\frac{2}{2}$ .

2 x = π n + \frac{π \cdot 2}{2 \cdot 2} - \frac{π}{4}

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

Step 2.1.3.2

Multiply

2

$2$ by

2

$2$ .

2 x = π n + \frac{π \cdot 2}{4} - \frac{π}{4}

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

2 x = π n + \frac{π \cdot 2}{4} - \frac{π}{4}

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

Step 2.1.4

Combine the numerators over the common denominator.

2 x = π n + \frac{π \cdot 2 - π}{4}

$2 x = π n + \frac{π \cdot 2 - π}{4}$

Step 2.1.5

Subtract

π

$π$ from

π \cdot 2

$π \cdot 2$ .

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Step 2.1.5.1

Reorder

π

$π$ and

2

$2$ .

2 x = π n + \frac{2 \cdot π - π}{4}

$2 x = π n + \frac{2 \cdot π - π}{4}$

Step 2.1.5.2

Subtract

π

$π$ from

2 \cdot π

$2 \cdot π$ .

2 x = π n + \frac{π}{4}

$2 x = π n + \frac{π}{4}$

2 x = π n + \frac{π}{4}

$2 x = π n + \frac{π}{4}$

2 x = π n + \frac{π}{4}

$2 x = π n + \frac{π}{4}$

Step 2.2

Divide each term in

2 x = π n + \frac{π}{4}

$2 x = π n + \frac{π}{4}$ by

2

$2$ and simplify.

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Step 2.2.1

Divide each term in

2 x = π n + \frac{π}{4}

$2 x = π n + \frac{π}{4}$ by

2

$2$ .

\frac{2 x}{2} = \frac{π n}{2} + \frac{\frac{π}{4}}{2}

$\frac{2 x}{2} = \frac{π n}{2} + \frac{\frac{π}{4}}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of

2

$2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π n}{2} + \frac{\frac{π}{4}}{2}$

Step 2.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{π n}{2} + \frac{\frac{π}{4}}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π n}{2} + \frac{π}{4} \cdot \frac{1}{2}$

Step 2.2.3.1.2

Multiply

$\frac{π}{4} \cdot \frac{1}{2}$ .

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Step 2.2.3.1.2.1

Multiply

$\frac{π}{4}$ by

$\frac{1}{2}$ .

$x = \frac{π n}{2} + \frac{π}{4 \cdot 2}$

Step 2.2.3.1.2.2

Multiply

$4$ by

$2$ .

$x = \frac{π n}{2} + \frac{π}{8}$

Step 3

The domain is all values of

$x$ that make the expression defined.

Set-Builder Notation:

${x | x \neq \frac{π n}{2} + \frac{π}{8}}$ , for any integer

$n$

Step 4