Trigonometry Examples

$\frac{12 \cot (x) - 5}{13 \sin (x) - 12} = 0$

Step 1

Set the numerator equal to zero.

$12 \cot (x) - 5 = 0$

Step 2

Solve the equation for $x$ .

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Step 2.1

Add $5$ to both sides of the equation.

$12 \cot (x) = 5$

Step 2.2

Divide each term in $12 \cot (x) = 5$ by $12$ and simplify.

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Step 2.2.1

Divide each term in $12 \cot (x) = 5$ by $12$ .

$\frac{12 \cot (x)}{12} = \frac{5}{12}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $12$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{12 \cot (x)}{12} = \frac{5}{12}$

Step 2.2.2.1.2

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

$\cot (x) = \frac{5}{12}$

Step 2.3

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

$x = arccot (\frac{5}{12})$

Step 2.4

Simplify the right side.

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Step 2.4.1

Evaluate $arccot (\frac{5}{12})$ .

$x = 1.1760052$

Step 2.5

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.

$x = (3.14159265) + 1.1760052$

Step 2.6

Solve for $x$ .

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Step 2.6.1

Remove parentheses.

$x = 3.14159265 + 1.1760052$

Step 2.6.2

Remove parentheses.

$x = (3.14159265) + 1.1760052$

Step 2.6.3

Add $3.14159265$ and $1.1760052$ .

$x = 4.31759786$

Step 2.7

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

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Step 2.7.1

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

$\frac{π}{| b |}$

Step 2.7.2

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

$\frac{π}{| 1 |}$

Step 2.7.3

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

$\frac{π}{1}$

Step 2.7.4

Divide $π$ by $1$ .

$π$

Step 2.8

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

$x = 1.1760052 + π n, 4.31759786 + π n$ , for any integer $n$

Step 3

Consolidate $1.1760052 + π n$ and $4.31759786 + π n$ to $1.1760052 + π n$ .

$x = 1.1760052 + π n$ , for any integer $n$

Step 4

Verify each of the solutions by substituting them into $\frac{12 \cot (x) - 5}{13 \sin (x) - 12} = 0$ and solving.

$x = 4.31759786 + 2 π n$ , for any integer $n$