Calculus Examples

$0 = \cos (x + \frac{π}{4})$

Step 1

Rewrite the equation as $\cos (x + \frac{π}{4}) = 0$ .

$\cos (x + \frac{π}{4}) = 0$

Step 2

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

$x + \frac{π}{4} = \arccos (0)$

Step 3

Simplify the right side.

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Step 3.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x + \frac{π}{4} = \frac{π}{2}$

Step 4

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1

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

$x = \frac{π}{2} - \frac{π}{4}$

Step 4.2

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

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

Step 4.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{π}{2}$ by $\frac{2}{2}$ .

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

Step 4.3.2

Multiply $2$ by $2$ .

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

Step 4.4

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{4}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{4}$

Step 4.5.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{4}$

Step 5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x + \frac{π}{4} = 2 π - \frac{π}{2}$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $2 π - \frac{π}{2}$ .

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Step 6.1.1

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

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

Step 6.1.2

Combine fractions.

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Step 6.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 6.1.2.2

Combine the numerators over the common denominator.

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

Step 6.1.3

Simplify the numerator.

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Step 6.1.3.1

Multiply $2$ by $2$ .

$x + \frac{π}{4} = \frac{4 π - π}{2}$

Step 6.1.3.2

Subtract $π$ from $4 π$ .

$x + \frac{π}{4} = \frac{3 π}{2}$

Step 6.2

Move all terms not containing $x$ to the right side of the equation.

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Step 6.2.1

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

$x = \frac{3 π}{2} - \frac{π}{4}$

Step 6.2.2

To write $\frac{3 π}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.3.1

Multiply $\frac{3 π}{2}$ by $\frac{2}{2}$ .

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

Step 6.2.3.2

Multiply $2$ by $2$ .

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

Step 6.2.4

Combine the numerators over the common denominator.

$x = \frac{3 π \cdot 2 - π}{4}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $2$ by $3$ .

$x = \frac{6 π - π}{4}$

Step 6.2.5.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{4}$

Step 7

Find the period of $\cos (x + \frac{π}{4})$ .

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| 1 |}$

Step 7.3

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

$\frac{2 π}{1}$

Step 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

The period of the $\cos (x + \frac{π}{4})$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

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

Step 9

Consolidate the answers.

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