Algebra Examples

$\cot (θ) = - 7$

Step 1

Solve for $θ$ .

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Step 1.1

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

$θ = arccot (- 7)$

Step 1.2

Simplify the right side.

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Step 1.2.1

Evaluate $arccot (- 7)$ .

$θ = 2.99969559$

Step 1.3

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.

$θ = 2.99969559 - (3.14159265)$

Step 1.4

Simplify the expression to find the second solution.

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Step 1.4.1

Add $2 π$ to $2.99969559 - (3.14159265)$ .

$θ = 2.99969559 - (3.14159265) + 2 π$

Step 1.4.2

The resulting angle of $6.14128825$ is positive and coterminal with $2.99969559 - (3.14159265)$ .

$θ = 6.14128825$

Step 1.5

Find the period of $\cot (θ)$ .

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Step 1.5.1

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

$\frac{π}{| b |}$

Step 1.5.2

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

$\frac{π}{| 1 |}$

Step 1.5.3

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

$\frac{π}{1}$

Step 1.5.4

Divide $π$ by $1$ .

$π$

Step 1.6

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

$θ = 2.99969559 + π n, 6.14128825 + π n$ , for any integer $n$

Step 1.7

Consolidate $2.99969559 + π n$ and $6.14128825 + π n$ to $2.99969559 + π n$ .

$θ = 2.99969559 + π n$ , for any integer $n$

Step 2

Apply the formula $\frac{y}{x} = \tan (θ)$ .

$\frac{y}{x} = \tan (2.99969559 + π n)$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides by $x$ .

$\frac{y}{x} x = \tan (2.99969559 + π n) x$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $x$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{y}{x} x = \tan (2.99969559 + π n) x$

Step 3.2.1.1.2

Rewrite the expression.

$y = \tan (2.99969559 + π n) x$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Reorder factors in $\tan (2.99969559 + π n) x$ .

$y = x \tan (2.99969559 + π n)$