Trigonometry Examples

$\frac{1}{2} = \cos (\frac{4}{3} \cdot (π x))$

Step 1

Rewrite the equation as $\cos (\frac{4}{3} \cdot (π x)) = \frac{1}{2}$ .

$\cos (\frac{4}{3} \cdot (π x)) = \frac{1}{2}$

Step 2

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

$\frac{4}{3} \cdot (π x) = \arccos (\frac{1}{2})$

Step 3

Simplify the left side.

Tap for more steps...

Step 3.1

Simplify $\frac{4}{3} \cdot (π x)$ .

Tap for more steps...

Step 3.1.1

Multiply $\frac{4}{3} (π x)$ .

Tap for more steps...

Step 3.1.1.1

Combine $π$ and $\frac{4}{3}$ .

$\frac{π \cdot 4}{3} x = \arccos (\frac{1}{2})$

Step 3.1.1.2

Combine $\frac{π \cdot 4}{3}$ and $x$ .

$\frac{π \cdot 4 x}{3} = \arccos (\frac{1}{2})$

Step 3.1.2

Move $4$ to the left of $π$ .

$\frac{4 π x}{3} = \arccos (\frac{1}{2})$

Step 4

Simplify the right side.

Tap for more steps...

Step 4.1

The exact value of $\arccos (\frac{1}{2})$ is $\frac{π}{3}$ .

$\frac{4 π x}{3} = \frac{π}{3}$

Step 5

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$4 π x = π$

Step 6

Divide each term in $4 π x = π$ by $4 π$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $4 π x = π$ by $4 π$ .

$\frac{4 π x}{4 π} = \frac{π}{4 π}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$\frac{4 π x}{4 π} = \frac{π}{4 π}$

Step 6.2.1.2

Rewrite the expression.

$\frac{π x}{π} = \frac{π}{4 π}$

Step 6.2.2

Cancel the common factor of $π$ .

Tap for more steps...

Step 6.2.2.1

Cancel the common factor.

$\frac{π x}{π} = \frac{π}{4 π}$

Step 6.2.2.2

Divide $x$ by $1$ .

$x = \frac{π}{4 π}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Cancel the common factor of $π$ .

Tap for more steps...

Step 6.3.1.1

Cancel the common factor.

$x = \frac{π}{4 π}$

Step 6.3.1.2

Rewrite the expression.

$x = \frac{1}{4}$

Step 7

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.

$\frac{4 π x}{3} = 2 π - \frac{π}{3}$

Step 8

Solve for $x$ .

Tap for more steps...

Step 8.1

Multiply both sides of the equation by $\frac{3}{4 π}$ .

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

Step 8.2

Simplify both sides of the equation.

Tap for more steps...

Step 8.2.1

Simplify the left side.

Tap for more steps...

Step 8.2.1.1

Simplify $\frac{3}{4 π} \cdot \frac{4 π x}{3}$ .

Tap for more steps...

Step 8.2.1.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.2.1.1.1.1

Cancel the common factor.

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

Step 8.2.1.1.1.2

Rewrite the expression.

$\frac{1}{4 π} (4 π x) = \frac{3}{4 π} (2 π - \frac{π}{3})$

Step 8.2.1.1.2

Cancel the common factor of $4 π$ .

Tap for more steps...

Step 8.2.1.1.2.1

Factor $4 π$ out of $4 π x$ .

$\frac{1}{4 π} (4 π (x)) = \frac{3}{4 π} (2 π - \frac{π}{3})$

Step 8.2.1.1.2.2

Cancel the common factor.

$\frac{1}{4 π} (4 π x) = \frac{3}{4 π} (2 π - \frac{π}{3})$

Step 8.2.1.1.2.3

Rewrite the expression.

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

Step 8.2.2

Simplify the right side.

Tap for more steps...

Step 8.2.2.1

Simplify $\frac{3}{4 π} (2 π - \frac{π}{3})$ .

Tap for more steps...

Step 8.2.2.1.1

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

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

Step 8.2.2.1.2

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 8.2.2.1.3

Combine the numerators over the common denominator.

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

Step 8.2.2.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.2.2.1.4.1

Cancel the common factor.

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

Step 8.2.2.1.4.2

Rewrite the expression.

$x = \frac{1}{4 π} (2 π \cdot 3 - π)$

Step 8.2.2.1.5

Multiply $3$ by $2$ .

$x = \frac{1}{4 π} (6 π - π)$

Step 8.2.2.1.6

Subtract $π$ from $6 π$ .

$x = \frac{1}{4 π} (5 π)$

Step 8.2.2.1.7

Cancel the common factor of $π$ .

Tap for more steps...

Step 8.2.2.1.7.1

Factor $π$ out of $4 π$ .

$x = \frac{1}{π \cdot 4} (5 π)$

Step 8.2.2.1.7.2

Factor $π$ out of $5 π$ .

$x = \frac{1}{π \cdot 4} (π \cdot 5)$

Step 8.2.2.1.7.3

Cancel the common factor.

$x = \frac{1}{π \cdot 4} (π \cdot 5)$

Step 8.2.2.1.7.4

Rewrite the expression.

$x = \frac{1}{4} \cdot 5$

Step 8.2.2.1.8

Combine $\frac{1}{4}$ and $5$ .

$x = \frac{5}{4}$

Step 9

Find the period of $\cos (\frac{4}{3} \cdot (π x))$ .

Tap for more steps...

Step 9.1

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

$\frac{2 π}{| b |}$

Step 9.2

Replace $b$ with $\frac{4}{3} \cdot π$ in the formula for period.

$\frac{2 π}{| \frac{4}{3} \cdot π |}$

Step 9.3

Simplify the denominator.

Tap for more steps...

Step 9.3.1

Combine $\frac{4}{3}$ and $π$ .

$\frac{2 π}{| \frac{4 π}{3} |}$

Step 9.3.2

$\frac{4 π}{3}$ is approximately $4.1887902$ which is positive so remove the absolute value

$\frac{2 π}{\frac{4 π}{3}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{4 π}$

Step 9.5

Cancel the common factor of $2 π$ .

Tap for more steps...

Step 9.5.1

Factor $2 π$ out of $4 π$ .

$2 π \frac{3}{2 π (2)}$

Step 9.5.2

Cancel the common factor.

$2 π \frac{3}{2 π \cdot 2}$

Step 9.5.3

Rewrite the expression.

$\frac{3}{2}$

Step 10

The period of the $\cos (\frac{4}{3} \cdot (π x))$ function is $\frac{3}{2}$ so values will repeat every $\frac{3}{2}$ radians in both directions.

$x = \frac{1}{4} + \frac{3 n}{2}, \frac{5}{4} + \frac{3 n}{2}$ , for any integer $n$