Trigonometry Examples

$\cos (π + x) + \cos (π + x) = 1$

Step 1

Add $\cos (π + x)$ and $\cos (π + x)$ .

$2 \cos (π + x) = 1$

Step 2

Divide each term in $2 \cos (π + x) = 1$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 \cos (π + x) = 1$ by $2$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $\cos (π + x)$ by $1$ .

$\cos (π + x) = \frac{1}{2}$

Step 3

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

$π + x = \arccos (\frac{1}{2})$

Step 4

Simplify the right side.

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Step 4.1

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

$π + x = \frac{π}{3}$

Step 5

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

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Step 5.1

Subtract $π$ from both sides of the equation.

$x = \frac{π}{3} - π$

Step 5.2

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

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

Step 5.3

Combine $- π$ and $\frac{3}{3}$ .

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

Step 5.4

Combine the numerators over the common denominator.

$x = \frac{π - π \cdot 3}{3}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Multiply $3$ by $- 1$ .

$x = \frac{π - 3 π}{3}$

Step 5.5.2

Subtract $3 π$ from $π$ .

$x = \frac{- 2 π}{3}$

Step 5.6

Move the negative in front of the fraction.

$x = - \frac{2 π}{3}$

Step 6

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.

$π + x = 2 π - \frac{π}{3}$

Step 7

Solve for $x$ .

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Step 7.1

Simplify $2 π - \frac{π}{3}$ .

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Step 7.1.1

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

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

Step 7.1.2

Combine fractions.

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Step 7.1.2.1

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 7.1.2.2

Combine the numerators over the common denominator.

$π + x = \frac{2 π \cdot 3 - π}{3}$

Step 7.1.3

Simplify the numerator.

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Step 7.1.3.1

Multiply $3$ by $2$ .

$π + x = \frac{6 π - π}{3}$

Step 7.1.3.2

Subtract $π$ from $6 π$ .

$π + x = \frac{5 π}{3}$

Step 7.2

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

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Step 7.2.1

Subtract $π$ from both sides of the equation.

$x = \frac{5 π}{3} - π$

Step 7.2.2

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

$x = \frac{5 π}{3} - π \cdot \frac{3}{3}$

Step 7.2.3

Combine $- π$ and $\frac{3}{3}$ .

$x = \frac{5 π}{3} + \frac{- π \cdot 3}{3}$

Step 7.2.4

Combine the numerators over the common denominator.

$x = \frac{5 π - π \cdot 3}{3}$

Step 7.2.5

Simplify the numerator.

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Step 7.2.5.1

Multiply $3$ by $- 1$ .

$x = \frac{5 π - 3 π}{3}$

Step 7.2.5.2

Subtract $3 π$ from $5 π$ .

$x = \frac{2 π}{3}$

Step 8

Find the period of $\cos (π + x)$ .

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 8.4

Divide $2 π$ by $1$ .

$2 π$

Step 9

Add $2 π$ to every negative angle to get positive angles.

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Step 9.1

Add $2 π$ to $- \frac{2 π}{3}$ to find the positive angle.

$- \frac{2 π}{3} + 2 π$

Step 9.2

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

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

Step 9.3

Combine fractions.

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Step 9.3.1

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 9.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - 2 π}{3}$

Step 9.4

Simplify the numerator.

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Step 9.4.1

Multiply $3$ by $2$ .

$\frac{6 π - 2 π}{3}$

Step 9.4.2

Subtract $2 π$ from $6 π$ .

$\frac{4 π}{3}$

Step 9.5

List the new angles.

$x = \frac{4 π}{3}$

Step 10

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

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