Finite Math Examples

$r = 4 \sin (θ)$ , $r = 4 \cos (θ)$

Step 1

Eliminate the equal sides of each equation and combine.

$4 \sin (θ) = 4 \cos (θ)$

Step 2

Solve $4 \sin (θ) = 4 \cos (θ)$ for $θ$ .

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Step 2.1

Divide each term in the equation by $\cos (θ)$ .

$\frac{4 \sin (θ)}{\cos (θ)} = \frac{4 \cos (θ)}{\cos (θ)}$

Step 2.2

Separate fractions.

$\frac{4}{1} \cdot \frac{\sin (θ)}{\cos (θ)} = \frac{4 \cos (θ)}{\cos (θ)}$

Step 2.3

Convert from $\frac{\sin (θ)}{\cos (θ)}$ to $\tan (θ)$ .

$\frac{4}{1} \cdot \tan (θ) = \frac{4 \cos (θ)}{\cos (θ)}$

Step 2.4

Divide $4$ by $1$ .

$4 \tan (θ) = \frac{4 \cos (θ)}{\cos (θ)}$

Step 2.5

Cancel the common factor of $\cos (θ)$ .

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Step 2.5.1

Cancel the common factor.

$4 \tan (θ) = \frac{4 \cos (θ)}{\cos (θ)}$

Step 2.5.2

Divide $4$ by $1$ .

$4 \tan (θ) = 4$

Step 2.6

Divide each term in $4 \tan (θ) = 4$ by $4$ and simplify.

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Step 2.6.1

Divide each term in $4 \tan (θ) = 4$ by $4$ .

$\frac{4 \tan (θ)}{4} = \frac{4}{4}$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Cancel the common factor of $4$ .

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Step 2.6.2.1.1

Cancel the common factor.

$\frac{4 \tan (θ)}{4} = \frac{4}{4}$

Step 2.6.2.1.2

Divide $\tan (θ)$ by $1$ .

$\tan (θ) = \frac{4}{4}$

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Divide $4$ by $4$ .

$\tan (θ) = 1$

Step 2.7

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

$θ = \arctan (1)$

Step 2.8

Simplify the right side.

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Step 2.8.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$θ = \frac{π}{4}$

Step 2.9

The tangent 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.

$θ = π + \frac{π}{4}$

Step 2.10

Simplify $π + \frac{π}{4}$ .

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Step 2.10.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$θ = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 2.10.2

Combine fractions.

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Step 2.10.2.1

Combine $π$ and $\frac{4}{4}$ .

$θ = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 2.10.2.2

Combine the numerators over the common denominator.

$θ = \frac{π \cdot 4 + π}{4}$

Step 2.10.3

Simplify the numerator.

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Step 2.10.3.1

Move $4$ to the left of $π$ .

$θ = \frac{4 \cdot π + π}{4}$

Step 2.10.3.2

Add $4 π$ and $π$ .

$θ = \frac{5 π}{4}$

Step 2.11

Find the period of $\tan (θ)$ .

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Step 2.11.1

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

$\frac{π}{| b |}$

Step 2.11.2

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

$\frac{π}{| 1 |}$

Step 2.11.3

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

$\frac{π}{1}$

Step 2.11.4

Divide $π$ by $1$ .

$π$

Step 2.12

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

$θ = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 3

Evaluate $r$ when $θ = \frac{π}{4} + π n$ .

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Step 3.1

Substitute $\frac{π}{4} + π n$ for $θ$ .

$r = 4 \cos (\frac{π}{4} + π n)$

Step 3.2

Remove parentheses.

$r = 4 \cos (\frac{π}{4} + π n)$

Step 4

Evaluate $r$ when $θ = \frac{5 π}{4} + π n$ .

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Step 4.1

Substitute $\frac{5 π}{4} + π n$ for $θ$ .

$r = 4 \cos (\frac{5 π}{4} + π n)$

Step 4.2

Remove parentheses.

$r = 4 \cos (\frac{5 π}{4} + π n)$

Step 5

The solution of the system of equations is all the values that make the system true.

$r = 4 \cos (\frac{π}{4} + π n), θ = \frac{π}{4} + π n r = 4 \cos (\frac{5 π}{4} + π n), θ = \frac{5 π}{4} + π n$

Step 6

List all of the solutions.

$r = 4 \cos (\frac{π}{4} + π n), θ = \frac{π}{4} + π n$

$r = 4 \cos (\frac{5 π}{4} + π n), θ = \frac{5 π}{4} + π n$