Trigonometry Examples

$\cos (x - \frac{π}{4}) - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

$\cos (x - \frac{π}{4}) = 1$

Step 2

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

$x - \frac{π}{4} = \arccos (1)$

Step 3

Simplify the right side.

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Step 3.1

The exact value of $\arccos (1)$ is $0$ .

$x - \frac{π}{4} = 0$

Step 4

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

$x = \frac{π}{4}$

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.

$x - \frac{π}{4} = 2 π - 0$

Step 6

Solve for $x$ .

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Step 6.1

Subtract $0$ from $2 π$ .

$x - \frac{π}{4} = 2 π$

Step 6.2

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

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Step 6.2.1

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

$x = 2 π + \frac{π}{4}$

Step 6.2.2

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

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

Step 6.2.3

Combine $2 π$ and $\frac{4}{4}$ .

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

Step 6.2.4

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 + π}{4}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $4$ by $2$ .

$x = \frac{8 π + π}{4}$

Step 6.2.5.2

Add $8 π$ and $π$ .

$x = \frac{9 π}{4}$

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.

$x = \frac{π}{4} + 2 π n, \frac{9 π}{4} + 2 π n$ , for any integer $n$

Step 9

Consolidate the answers.

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