Trigonometry Examples

$\cos (x + \frac{π}{4}) = - \frac{\sqrt{2}}{2}$

Step 1

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

$x + \frac{π}{4} = \arccos (- \frac{\sqrt{2}}{2})$

Step 2

Simplify the right side.

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Step 2.1

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

$x + \frac{π}{4} = \frac{3 π}{4}$

Step 3

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

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Step 3.1

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

$x = \frac{3 π}{4} - \frac{π}{4}$

Step 3.2

Combine the numerators over the common denominator.

$x = \frac{3 π - π}{4}$

Step 3.3

Subtract $π$ from $3 π$ .

$x = \frac{2 π}{4}$

Step 3.4

Cancel the common factor of $2$ and $4$ .

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Step 3.4.1

Factor $2$ out of $2 π$ .

$x = \frac{2 (π)}{4}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $4$ .

$x = \frac{2 π}{2 \cdot 2}$

Step 3.4.2.2

Cancel the common factor.

$x = \frac{2 π}{2 \cdot 2}$

Step 3.4.2.3

Rewrite the expression.

$x = \frac{π}{2}$

Step 4

The cosine 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.

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

Step 5

Solve for $x$ .

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Step 5.1

Simplify $2 π - \frac{3 π}{4}$ .

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Step 5.1.1

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

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

Step 5.1.2

Combine fractions.

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Step 5.1.2.1

Combine $2 π$ and $\frac{4}{4}$ .

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

Step 5.1.2.2

Combine the numerators over the common denominator.

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

Step 5.1.3

Simplify the numerator.

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Step 5.1.3.1

Multiply $4$ by $2$ .

$x + \frac{π}{4} = \frac{8 π - 3 π}{4}$

Step 5.1.3.2

Subtract $3 π$ from $8 π$ .

$x + \frac{π}{4} = \frac{5 π}{4}$

Step 5.2

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

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Step 5.2.1

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

$x = \frac{5 π}{4} - \frac{π}{4}$

Step 5.2.2

Combine the numerators over the common denominator.

$x = \frac{5 π - π}{4}$

Step 5.2.3

Subtract $π$ from $5 π$ .

$x = \frac{4 π}{4}$

Step 5.2.4

Cancel the common factor of $4$ .

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Step 5.2.4.1

Cancel the common factor.

$x = \frac{4 π}{4}$

Step 5.2.4.2

Divide $π$ by $1$ .

$x = π$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 1 |}$

Step 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 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

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