Trigonometry Examples

$7 \cot^{2} (t) + 1.2 = 6$

Step 1

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

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Step 1.1

Subtract $1.2$ from both sides of the equation.

$7 \cot^{2} (t) = 6 - 1.2$

Step 1.2

Subtract $1.2$ from $6$ .

$7 \cot^{2} (t) = 4.8$

Step 2

Divide each term in $7 \cot^{2} (t) = 4.8$ by $7$ and simplify.

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Step 2.1

Divide each term in $7 \cot^{2} (t) = 4.8$ by $7$ .

$\frac{7 \cot^{2} (t)}{7} = \frac{4.8}{7}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $7$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{7 \cot^{2} (t)}{7} = \frac{4.8}{7}$

Step 2.2.1.2

Divide $\cot^{2} (t)$ by $1$ .

$\cot^{2} (t) = \frac{4.8}{7}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $4.8$ by $7$ .

$\cot^{2} (t) = 0.6\overline{857142}$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\cot (t) = \pm \sqrt{0.6\overline{857142}}$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$\cot (t) = \sqrt{0.6\overline{857142}}$

Step 4.2

Next, use the negative value of the $\pm$ to find the second solution.

$\cot (t) = - \sqrt{0.6\overline{857142}}$

Step 4.3

The complete solution is the result of both the positive and negative portions of the solution.

$\cot (t) = \sqrt{0.6\overline{857142}}, - \sqrt{0.6\overline{857142}}$

Step 5

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

$\cot (t) = \sqrt{0.6\overline{857142}}$

$\cot (t) = - \sqrt{0.6\overline{857142}}$

Step 6

Solve for $t$ in $\cot (t) = \sqrt{0.6\overline{857142}}$ .

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Step 6.1

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

$t = arccot (\sqrt{0.6\overline{857142}})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Evaluate $arccot (\sqrt{0.6\overline{857142}})$ .

$t = 0.87916718$

Step 6.3

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.

$t = (3.14159265) + 0.87916718$

Step 6.4

Solve for $t$ .

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Step 6.4.1

Remove parentheses.

$t = 3.14159265 + 0.87916718$

Step 6.4.2

Remove parentheses.

$t = (3.14159265) + 0.87916718$

Step 6.4.3

Add $3.14159265$ and $0.87916718$ .

$t = 4.02075984$

Step 6.5

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

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Step 6.5.1

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

$\frac{π}{| b |}$

Step 6.5.2

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

$\frac{π}{| 1 |}$

Step 6.5.3

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

$\frac{π}{1}$

Step 6.5.4

Divide $π$ by $1$ .

$π$

Step 6.6

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

$t = 0.87916718 + π n, 4.02075984 + π n$ , for any integer $n$

Step 7

Solve for $t$ in $\cot (t) = - \sqrt{0.6\overline{857142}}$ .

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Step 7.1

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

$t = arccot (- \sqrt{0.6\overline{857142}})$

Step 7.2

Simplify the right side.

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Step 7.2.1

Evaluate $arccot (- \sqrt{0.6\overline{857142}})$ .

$t = 2.26242546$

Step 7.3

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.

$t = 2.26242546 - (3.14159265)$

Step 7.4

Simplify the expression to find the second solution.

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Step 7.4.1

Add $2 π$ to $2.26242546 - (3.14159265)$ .

$t = 2.26242546 - (3.14159265) + 2 π$

Step 7.4.2

The resulting angle of $5.40401811$ is positive and coterminal with $2.26242546 - (3.14159265)$ .

$t = 5.40401811$

Step 7.5

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

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Step 7.5.1

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

$\frac{π}{| b |}$

Step 7.5.2

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

$\frac{π}{| 1 |}$

Step 7.5.3

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

$\frac{π}{1}$

Step 7.5.4

Divide $π$ by $1$ .

$π$

Step 7.6

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

$t = 2.26242546 + π n, 5.40401811 + π n$ , for any integer $n$

Step 8

List all of the solutions.

$t = 0.87916718 + π n, 4.02075984 + π n, 2.26242546 + π n, 5.40401811 + π n$ , for any integer $n$

Step 9

Consolidate the solutions.

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Step 9.1

Consolidate $0.87916718 + π n$ and $4.02075984 + π n$ to $0.87916718 + π n$ .

$t = 0.87916718 + π n, 2.26242546 + π n, 5.40401811 + π n$ , for any integer $n$

Step 9.2

Consolidate $2.26242546 + π n$ and $5.40401811 + π n$ to $2.26242546 + π n$ .

$t = 0.87916718 + π n, 2.26242546 + π n$ , for any integer $n$