Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{x}{3} = \frac{π}{3}$

Step 3

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

$x = π$

Step 4

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.

$\frac{x}{3} = 2 π - \frac{π}{3}$

Step 5

Solve for $x$ .

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Step 5.1

Multiply both sides of the equation by $3$ .

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

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Cancel the common factor of $3$ .

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Step 5.2.1.1.1

Cancel the common factor.

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

Step 5.2.1.1.2

Rewrite the expression.

$x = 3 (2 π - \frac{π}{3})$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $3 (2 π - \frac{π}{3})$ .

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Step 5.2.2.1.1

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

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

Step 5.2.2.1.2

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 5.2.2.1.3

Combine the numerators over the common denominator.

$x = 3 \frac{2 π \cdot 3 - π}{3}$

Step 5.2.2.1.4

Cancel the common factor of $3$ .

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Step 5.2.2.1.4.1

Cancel the common factor.

$x = 3 \frac{2 π \cdot 3 - π}{3}$

Step 5.2.2.1.4.2

Rewrite the expression.

$x = 2 π \cdot 3 - π$

Step 5.2.2.1.5

Multiply $3$ by $2$ .

$x = 6 π - π$

Step 5.2.2.1.6

Subtract $π$ from $6 π$ .

$x = 5 π$

Step 6

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

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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 $\frac{1}{3}$ in the formula for period.

$\frac{2 π}{| \frac{1}{3} |}$

Step 6.3

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{3}}$

Step 6.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 3$

Step 6.5

Multiply $3$ by $2$ .

$6 π$

Step 7

The period of the $\cos (\frac{x}{3})$ function is $6 π$ so values will repeat every $6 π$ radians in both directions.

$x = π + 6 π n, 5 π + 6 π n$ , for any integer $n$