Trigonometry Examples

$5 \cot (x) + 1 = 4$

Step 1

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

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Step 1.1

Subtract $1$ from both sides of the equation.

$5 \cot (x) = 4 - 1$

Step 1.2

Subtract $1$ from $4$ .

$5 \cot (x) = 3$

Step 2

Divide each term in $5 \cot (x) = 3$ by $5$ and simplify.

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Step 2.1

Divide each term in $5 \cot (x) = 3$ by $5$ .

$\frac{5 \cot (x)}{5} = \frac{3}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{5 \cot (x)}{5} = \frac{3}{5}$

Step 2.2.1.2

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

$\cot (x) = \frac{3}{5}$

Step 3

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

$x = arccot (\frac{3}{5})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $arccot (\frac{3}{5})$ .

$x = 1.03037682$

Step 5

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

$x = (3.14159265) + 1.03037682$

Step 6

Solve for $x$ .

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Step 6.1

Remove parentheses.

$x = 3.14159265 + 1.03037682$

Step 6.2

Remove parentheses.

$x = (3.14159265) + 1.03037682$

Step 6.3

Add $3.14159265$ and $1.03037682$ .

$x = 4.17196948$

Step 7

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

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Step 7.1

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

$\frac{π}{| b |}$

Step 7.2

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

$\frac{π}{| 1 |}$

Step 7.3

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

$\frac{π}{1}$

Step 7.4

Divide $π$ by $1$ .

$π$

Step 8

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

$x = 1.03037682 + π n, 4.17196948 + π n$ , for any integer $n$

Step 9

Consolidate $1.03037682 + π n$ and $4.17196948 + π n$ to $1.03037682 + π n$ .

$x = 1.03037682 + π n$ , for any integer $n$