Calculus Examples

$r^{2} (\cos^{2} (x)) + r^{2} (\sin^{2} (x)) - 2 \sin (x) = 9$

Step 1

Simplify the left side.

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Step 1.1

Simplify $r^{2} \cos^{2} (x) + r^{2} \sin^{2} (x) - 2 \sin (x)$ .

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Step 1.1.1

Factor $r^{2}$ out of $r^{2} \cos^{2} (x)$ .

$r^{2} (\cos^{2} (x)) + r^{2} \sin^{2} (x) - 2 \sin (x) = 9$

Step 1.1.2

Factor $r^{2}$ out of $r^{2} \sin^{2} (x)$ .

$r^{2} (\cos^{2} (x)) + r^{2} (\sin^{2} (x)) - 2 \sin (x) = 9$

Step 1.1.3

Factor $r^{2}$ out of $r^{2} (\cos^{2} (x)) + r^{2} (\sin^{2} (x))$ .

$r^{2} (\cos^{2} (x) + \sin^{2} (x)) - 2 \sin (x) = 9$

Step 1.1.4

Rearrange terms.

$r^{2} (\sin^{2} (x) + \cos^{2} (x)) - 2 \sin (x) = 9$

Step 1.1.5

Apply pythagorean identity.

$r^{2} \cdot 1 - 2 \sin (x) = 9$

Step 1.1.6

Multiply $r^{2}$ by $1$ .

$r^{2} - 2 \sin (x) = 9$

Step 2

Add $2 \sin (x)$ to both sides of the equation.

$r^{2} = 9 + 2 \sin (x)$

Step 3

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

$r = \pm \sqrt{9 + 2 \sin (x)}$

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.

$r = \sqrt{9 + 2 \sin (x)}$

Step 4.2

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

$r = - \sqrt{9 + 2 \sin (x)}$

Step 4.3

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

$r = \sqrt{9 + 2 \sin (x)}$

$r = - \sqrt{9 + 2 \sin (x)}$

$r = \sqrt{9 + 2 \sin (x)}$

$r = - \sqrt{9 + 2 \sin (x)}$