Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

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

$2 x + \frac{π}{4} = \frac{π}{4}$

Step 3

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

Tap for more steps...

Step 3.1

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

$2 x = \frac{π}{4} - \frac{π}{4}$

Step 3.2

Combine the numerators over the common denominator.

$2 x = \frac{π - π}{4}$

Step 3.3

Subtract $π$ from $π$ .

$2 x = \frac{0}{4}$

Step 3.4

Divide $0$ by $4$ .

$2 x = 0$

Step 4

Divide each term in $2 x = 0$ by $2$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Divide $0$ by $2$ .

$x = 0$

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.

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

Step 6

Solve for $x$ .

Tap for more steps...

Step 6.1

Simplify $2 π - \frac{π}{4}$ .

Tap for more steps...

Step 6.1.1

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

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

Step 6.1.2

Combine fractions.

Tap for more steps...

Step 6.1.2.1

Combine $2 π$ and $\frac{4}{4}$ .

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

Step 6.1.2.2

Combine the numerators over the common denominator.

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

Step 6.1.3

Simplify the numerator.

Tap for more steps...

Step 6.1.3.1

Multiply $4$ by $2$ .

$2 x + \frac{π}{4} = \frac{8 π - π}{4}$

Step 6.1.3.2

Subtract $π$ from $8 π$ .

$2 x + \frac{π}{4} = \frac{7 π}{4}$

Step 6.2

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

Tap for more steps...

Step 6.2.1

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

$2 x = \frac{7 π}{4} - \frac{π}{4}$

Step 6.2.2

Combine the numerators over the common denominator.

$2 x = \frac{7 π - π}{4}$

Step 6.2.3

Subtract $π$ from $7 π$ .

$2 x = \frac{6 π}{4}$

Step 6.2.4

Cancel the common factor of $6$ and $4$ .

Tap for more steps...

Step 6.2.4.1

Factor $2$ out of $6 π$ .

$2 x = \frac{2 (3 π)}{4}$

Step 6.2.4.2

Cancel the common factors.

Tap for more steps...

Step 6.2.4.2.1

Factor $2$ out of $4$ .

$2 x = \frac{2 (3 π)}{2 (2)}$

Step 6.2.4.2.2

Cancel the common factor.

$2 x = \frac{2 (3 π)}{2 \cdot 2}$

Step 6.2.4.2.3

Rewrite the expression.

$2 x = \frac{3 π}{2}$

Step 6.3

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ and simplify.

Tap for more steps...

Step 6.3.1

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2}$

Step 6.3.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{2}$

Step 6.3.3

Simplify the right side.

Tap for more steps...

Step 6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 6.3.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 6.3.3.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{2}$ .

$x = \frac{3 π}{2 \cdot 2}$

Step 6.3.3.2.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 7

Find the period of $\cos (2 x + \frac{π}{4})$ .

Tap for more steps...

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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 7.4.2

Divide $π$ by $1$ .

$π$

Step 8

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

$x = π n, \frac{3 π}{4} + π n$ , for any integer $n$