Trigonometry Examples

$\cos^{2} (x) + \sin^{2} (x) = {(\frac{x}{r})}^{2} + {(\frac{y}{r})}^{2}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract ${(\frac{x}{r})}^{2}$ from both sides of the equation.

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

Step 1.2

Subtract ${(\frac{y}{r})}^{2}$ from both sides of the equation.

$\cos^{2} (x) + \sin^{2} (x) - {(\frac{x}{r})}^{2} - {(\frac{y}{r})}^{2} = 0$

Step 2

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

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Step 2.1

Rearrange terms.

$\sin^{2} (x) + \cos^{2} (x) - {(\frac{x}{r})}^{2} - {(\frac{y}{r})}^{2} = 0$

Step 2.2

Apply pythagorean identity.

$1 - {(\frac{x}{r})}^{2} - {(\frac{y}{r})}^{2} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the product rule to $\frac{x}{r}$ .

$1 - \frac{x^{2}}{r^{2}} - {(\frac{y}{r})}^{2} = 0$

Step 2.3.2

Apply the product rule to $\frac{y}{r}$ .

$1 - \frac{x^{2}}{r^{2}} - \frac{y^{2}}{r^{2}} = 0$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1.1

Subtract $1$ from both sides of the equation.

$- \frac{x^{2}}{r^{2}} - \frac{y^{2}}{r^{2}} = - 1$

Step 3.1.2

Add $\frac{y^{2}}{r^{2}}$ to both sides of the equation.

$- \frac{x^{2}}{r^{2}} = - 1 + \frac{y^{2}}{r^{2}}$

Step 3.2

Divide each term in $- \frac{x^{2}}{r^{2}} = - 1 + \frac{y^{2}}{r^{2}}$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- \frac{x^{2}}{r^{2}} = - 1 + \frac{y^{2}}{r^{2}}$ by $- 1$ .

$\frac{- \frac{x^{2}}{r^{2}}}{- 1} = \frac{- 1}{- 1} + \frac{\frac{y^{2}}{r^{2}}}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x^{2}}{r^{2}}}{1} = \frac{- 1}{- 1} + \frac{\frac{y^{2}}{r^{2}}}{- 1}$

Step 3.2.2.2

Divide $\frac{x^{2}}{r^{2}}$ by $1$ .

$\frac{x^{2}}{r^{2}} = \frac{- 1}{- 1} + \frac{\frac{y^{2}}{r^{2}}}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Divide $- 1$ by $- 1$ .

$\frac{x^{2}}{r^{2}} = 1 + \frac{\frac{y^{2}}{r^{2}}}{- 1}$

Step 3.2.3.1.2

Move the negative one from the denominator of $\frac{\frac{y^{2}}{r^{2}}}{- 1}$ .

$\frac{x^{2}}{r^{2}} = 1 - 1 \cdot \frac{y^{2}}{r^{2}}$

Step 3.2.3.1.3

Rewrite $- 1 \cdot \frac{y^{2}}{r^{2}}$ as $- \frac{y^{2}}{r^{2}}$ .

$\frac{x^{2}}{r^{2}} = 1 - \frac{y^{2}}{r^{2}}$

Step 3.3

Multiply both sides by $r^{2}$ .

$\frac{x^{2}}{r^{2}} r^{2} = (1 - \frac{y^{2}}{r^{2}}) r^{2}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Cancel the common factor of $r^{2}$ .

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Step 3.4.1.1.1

Cancel the common factor.

$\frac{x^{2}}{r^{2}} r^{2} = (1 - \frac{y^{2}}{r^{2}}) r^{2}$

Step 3.4.1.1.2

Rewrite the expression.

$x^{2} = (1 - \frac{y^{2}}{r^{2}}) r^{2}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $(1 - \frac{y^{2}}{r^{2}}) r^{2}$ .

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Step 3.4.2.1.1

Apply the distributive property.

$x^{2} = 1 r^{2} - \frac{y^{2}}{r^{2}} r^{2}$

Step 3.4.2.1.2

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

$x^{2} = r^{2} - \frac{y^{2}}{r^{2}} r^{2}$

Step 3.4.2.1.3

Cancel the common factor of $r^{2}$ .

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Step 3.4.2.1.3.1

Move the leading negative in $- \frac{y^{2}}{r^{2}}$ into the numerator.

$x^{2} = r^{2} + \frac{- y^{2}}{r^{2}} r^{2}$

Step 3.4.2.1.3.2

Cancel the common factor.

$x^{2} = r^{2} + \frac{- y^{2}}{r^{2}} r^{2}$

Step 3.4.2.1.3.3

Rewrite the expression.

$x^{2} = r^{2} - y^{2}$

Step 3.5

Solve for $x$ .

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Step 3.5.1

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

$x = \pm \sqrt{r^{2} - y^{2}}$

Step 3.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = r$ and $b = y$ .

$x = \pm \sqrt{(r + y) (r - y)}$

Step 3.5.3

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

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Step 3.5.3.1

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

$x = \sqrt{(r + y) (r - y)}$

Step 3.5.3.2

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