Precalculus Examples

$\cos (x + \frac{π}{4}) - \cos (x - \frac{π}{4}) = 1$ , $[0, 2 π)$

Step 1

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = \frac{5 π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 2

Find the values of $n$ that produce a value within the interval $[0, 2 π)$ .

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Step 2.1

Plug in $0$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 2.1.1

Plug in $0$ for $n$ .

$\frac{5 π}{4} + 2 π (0)$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Multiply $2 π (0)$ .

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Step 2.1.2.1.1

Multiply $0$ by $2$ .

$\frac{5 π}{4} + 0 π$

Step 2.1.2.1.2

Multiply $0$ by $π$ .

$\frac{5 π}{4} + 0$

Step 2.1.2.2

Add $\frac{5 π}{4}$ and $0$ .

$\frac{5 π}{4}$

Step 2.1.3

The interval $[0, 2 π)$ contains $\frac{5 π}{4}$ .

$x = \frac{5 π}{4}$

Step 2.2

Plug in $0$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 2.2.1

Plug in $0$ for $n$ .

$\frac{7 π}{4} + 2 π (0)$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Multiply $2 π (0)$ .

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Step 2.2.2.1.1

Multiply $0$ by $2$ .

$\frac{7 π}{4} + 0 π$

Step 2.2.2.1.2

Multiply $0$ by $π$ .

$\frac{7 π}{4} + 0$

Step 2.2.2.2

Add $\frac{7 π}{4}$ and $0$ .

$\frac{7 π}{4}$

Step 2.2.3

The interval $[0, 2 π)$ contains $\frac{7 π}{4}$ .

$x = \frac{5 π}{4}, \frac{7 π}{4}$

Step 3