Trigonometry Examples

$\csc^{2} (x) + 0.5 \cot (x) - 5 = 0$

Step 1

Replace the $\csc^{2} (x)$ with $\cot^{2} (x) + 1$ based on the $\cot^{2} (x) + 1 = \csc^{2} (x)$ identity.

$(\cot^{2} (x) + 1) + 0.5 \cot (x) - 5 = 0$

Step 2

Subtract $5$ from $1$ .

$\cot^{2} (x) + 0.5 \cot (x) - 4 = 0$

Step 3

Substitute $u$ for $\cot (x)$ .

${(u)}^{2} + 0.5 (u) - 4 = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = 1$ , $b = 0.5$ , and $c = - 4$ into the quadratic formula and solve for $u$ .

$\frac{- 0.5 \pm \sqrt{{0.5}^{2} - 4 \cdot (1 \cdot - 4)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $0.5$ to the power of $2$ .

$u = \frac{- 0.5 \pm \sqrt{0.25 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 0.5 \pm \sqrt{0.25 - 4 \cdot - 4}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 4$ .

$u = \frac{- 0.5 \pm \sqrt{0.25 + 16}}{2 \cdot 1}$

Step 6.1.3

Add $0.25$ and $16$ .

$u = \frac{- 0.5 \pm \sqrt{16.25}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$u = \frac{- 0.5 \pm \sqrt{16.25}}{2}$

Step 7

The final answer is the combination of both solutions.

$u = 1.76556443, - 2.26556443$

Step 8

Substitute $\cot (x)$ for $u$ .

$\cot (x) = 1.76556443, - 2.26556443$

Step 9

Set up each of the solutions to solve for $x$ .

$\cot (x) = 1.76556443$

$\cot (x) = - 2.26556443$

Step 10

Solve for $x$ in $\cot (x) = 1.76556443$ .

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Step 10.1

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

$x = arccot (1.76556443)$

Step 10.2

Simplify the right side.

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Step 10.2.1

Evaluate $arccot (1.76556443)$ .

$x = 0.5153404$

Step 10.3

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.

$x = (3.14159265) + 0.5153404$

Step 10.4

Solve for $x$ .

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Step 10.4.1

Remove parentheses.

$x = 3.14159265 + 0.5153404$

Step 10.4.2

Remove parentheses.

$x = (3.14159265) + 0.5153404$

Step 10.4.3

Add $3.14159265$ and $0.5153404$ .

$x = 3.65693305$

Step 10.5

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

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Step 10.5.1

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

$\frac{π}{| b |}$

Step 10.5.2

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

$\frac{π}{| 1 |}$

Step 10.5.3

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

$\frac{π}{1}$

Step 10.5.4

Divide $π$ by $1$ .

$π$

Step 10.6

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

$x = 0.5153404 + π n, 3.65693305 + π n$ , for any integer $n$

Step 11

Solve for $x$ in $\cot (x) = - 2.26556443$ .

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Step 11.1

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

$x = arccot (- 2.26556443)$

Step 11.2

Simplify the right side.

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Step 11.2.1

Evaluate $arccot (- 2.26556443)$ .

$x = 2.7259209$

Step 11.3

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.

$x = 2.7259209 - (3.14159265)$

Step 11.4

Simplify the expression to find the second solution.

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Step 11.4.1

Add $2 π$ to $2.7259209 - (3.14159265)$ .

$x = 2.7259209 - (3.14159265) + 2 π$

Step 11.4.2

The resulting angle of $5.86751355$ is positive and coterminal with $2.7259209 - (3.14159265)$ .

$x = 5.86751355$

Step 11.5

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

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Step 11.5.1

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

$\frac{π}{| b |}$

Step 11.5.2

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

$\frac{π}{| 1 |}$

Step 11.5.3

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

$\frac{π}{1}$

Step 11.5.4

Divide $π$ by $1$ .

$π$

Step 11.6

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

$x = 2.7259209 + π n, 5.86751355 + π n$ , for any integer $n$

Step 12

List all of the solutions.

$x = 0.5153404 + π n, 3.65693305 + π n, 2.7259209 + π n, 5.86751355 + π n$ , for any integer $n$

Step 13

Consolidate the solutions.

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Step 13.1

Consolidate $0.5153404 + π n$ and $3.65693305 + π n$ to $0.5153404 + π n$ .

$x = 0.5153404 + π n, 2.7259209 + π n, 5.86751355 + π n$ , for any integer $n$

Step 13.2

Consolidate $2.7259209 + π n$ and $5.86751355 + π n$ to $2.7259209 + π n$ .

$x = 0.5153404 + π n, 2.7259209 + π n$ , for any integer $n$