Trigonometry Examples

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

Step 1

Move all terms not containing $\cos (x)$ to the right side of the equation.

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Step 1.1

Subtract $1$ from both sides of the equation.

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

Step 1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.3

Combine $- 1$ and $\frac{2}{2}$ .

$- \cos (x) = \frac{1}{2} + \frac{- 1 \cdot 2}{2}$

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 1$ by $2$ .

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

Step 1.5.2

Subtract $2$ from $1$ .

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

Step 1.6

Move the negative in front of the fraction.

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

Step 2

Divide each term in $- \cos (x) = - \frac{1}{2}$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- \cos (x) = - \frac{1}{2}$ by $- 1$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2

Divide $\cos (x)$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Dividing two negative values results in a positive value.

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

Step 2.3.2

Divide $\frac{1}{2}$ by $1$ .

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

Step 3

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

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

Step 4

Simplify the right side.

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Step 4.1

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

$x = \frac{π}{3}$

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.

$x = 2 π - \frac{π}{3}$

Step 6

Simplify $2 π - \frac{π}{3}$ .

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Step 6.1

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

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

Step 6.2

Combine fractions.

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Step 6.2.1

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - π}{3}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - π}{3}$

Step 6.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{3}$

Step 7

Find the period of $\cos (x)$ .

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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 $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

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