Trigonometry Examples

$5 \cot (x) + 12 = 6 \cot (x) + 13$

Step 1

Move all terms containing $\cot (x)$ to the left side of the equation.

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Step 1.1

Subtract $6 \cot (x)$ from both sides of the equation.

$5 \cot (x) + 12 - 6 \cot (x) = 13$

Step 1.2

Subtract $6 \cot (x)$ from $5 \cot (x)$ .

$- \cot (x) + 12 = 13$

Step 2

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

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Step 2.1

Subtract $12$ from both sides of the equation.

$- \cot (x) = 13 - 12$

Step 2.2

Subtract $12$ from $13$ .

$- \cot (x) = 1$

Step 3

Divide each term in $- \cot (x) = 1$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- \cot (x) = 1$ by $- 1$ .

$\frac{- \cot (x)}{- 1} = \frac{1}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{\cot (x)}{1} = \frac{1}{- 1}$

Step 3.2.2

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

$\cot (x) = \frac{1}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $1$ by $- 1$ .

$\cot (x) = - 1$

Step 4

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

$x = arccot (- 1)$

Step 5

Simplify the right side.

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Step 5.1

The exact value of $arccot (- 1)$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 6

The cotangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$x = \frac{3 π}{4} - π$

Step 7

Simplify the expression to find the second solution.

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Step 7.1

Add $2 π$ to $\frac{3 π}{4} - π$ .

$x = \frac{3 π}{4} - π + 2 π$

Step 7.2

The resulting angle of $\frac{7 π}{4}$ is positive and coterminal with $\frac{3 π}{4} - π$ .

$x = \frac{7 π}{4}$

Step 8

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

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Step 8.1

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

$\frac{π}{| b |}$

Step 8.2

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

$\frac{π}{| 1 |}$

Step 8.3

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

$\frac{π}{1}$

Step 8.4

Divide $π$ by $1$ .

$π$

Step 9

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

$x = \frac{3 π}{4} + π n, \frac{7 π}{4} + π n$ , for any integer $n$

Step 10

Consolidate the answers.

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