Trigonometry Examples

$f (x) = - \sqrt{49 - x^{2}}$

Step 1

Write $f (x) = - \sqrt{49 - x^{2}}$ as an equation.

$y = - \sqrt{49 - x^{2}}$

Step 2

Interchange the variables.

$x = - \sqrt{49 - y^{2}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- \sqrt{49 - y^{2}} = x$ .

$- \sqrt{49 - y^{2}} = x$

Step 3.2

Divide each term in $- \sqrt{49 - y^{2}} = x$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- \sqrt{49 - y^{2}} = x$ by $- 1$ .

$\frac{- \sqrt{49 - y^{2}}}{- 1} = \frac{x}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Reduce the expression by cancelling the common factors.

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Step 3.2.2.1.1

Dividing two negative values results in a positive value.

$\frac{\sqrt{49 - y^{2}}}{1} = \frac{x}{- 1}$

Step 3.2.2.1.2

Write the expression using exponents.

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Step 3.2.2.1.2.1

Divide $\sqrt{49 - y^{2}}$ by $1$ .

$\sqrt{49 - y^{2}} = \frac{x}{- 1}$

Step 3.2.2.1.2.2

Rewrite $49$ as $7^{2}$ .

$\sqrt{7^{2} - y^{2}} = \frac{x}{- 1}$

Step 3.2.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 7$ and $b = y$ .

$\sqrt{(7 + y) (7 - y)} = \frac{x}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$\sqrt{(7 + y) (7 - y)} = - 1 \cdot x$

Step 3.2.3.2

Rewrite $- 1 \cdot x$ as $- x$ .

$\sqrt{(7 + y) (7 - y)} = - x$

Step 3.3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{(7 + y) (7 - y)}}^{2} = {(- x)}^{2}$

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(7 + y) (7 - y)}$ as ${((7 + y) (7 - y))}^{\frac{1}{2}}$ .

${({((7 + y) (7 - y))}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify ${({((7 + y) (7 - y))}^{\frac{1}{2}})}^{2}$ .

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Step 3.4.2.1.1

Multiply the exponents in ${({((7 + y) (7 - y))}^{\frac{1}{2}})}^{2}$ .

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Step 3.4.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${((7 + y) (7 - y))}^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Step 3.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

${((7 + y) (7 - y))}^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

${((7 + y) (7 - y))}^{1} = {(- x)}^{2}$

Step 3.4.2.1.2

Expand $(7 + y) (7 - y)$ using the FOIL Method.

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Step 3.4.2.1.2.1

Apply the distributive property.

${(7 (7 - y) + y (7 - y))}^{1} = {(- x)}^{2}$

Step 3.4.2.1.2.2

Apply the distributive property.

${(7 \cdot 7 + 7 (- y) + y (7 - y))}^{1} = {(- x)}^{2}$

Step 3.4.2.1.2.3

Apply the distributive property.

${(7 \cdot 7 + 7 (- y) + y \cdot 7 + y (- y))}^{1} = {(- x)}^{2}$

Step 3.4.2.1.3

Simplify and combine like terms.

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Step 3.4.2.1.3.1

Simplify each term.

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Step 3.4.2.1.3.1.1

Multiply $7$ by $7$ .

${(49 + 7 (- y) + y \cdot 7 + y (- y))}^{1} = {(- x)}^{2}$

Step 3.4.2.1.3.1.2

Multiply $- 1$ by $7$ .

${(49 - 7 y + y \cdot 7 + y (- y))}^{1} = {(- x)}^{2}$

Step 3.4.2.1.3.1.3

Move $7$ to the left of $y$ .

${(49 - 7 y + 7 \cdot y + y (- y))}^{1} = {(- x)}^{2}$

Step 3.4.2.1.3.1.4

Rewrite using the commutative property of multiplication.

${(49 - 7 y + 7 y - y \cdot y)}^{1} = {(- x)}^{2}$

Step 3.4.2.1.3.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 3.4.2.1.3.1.5.1

Move $y$ .

${(49 - 7 y + 7 y - (y \cdot y))}^{1} = {(- x)}^{2}$

Step 3.4.2.1.3.1.5.2

Multiply $y$ by $y$ .

${(49 - 7 y + 7 y - y^{2})}^{1} = {(- x)}^{2}$

Step 3.4.2.1.3.2

Add $- 7 y$ and $7 y$ .

${(49 + 0 - y^{2})}^{1} = {(- x)}^{2}$

Step 3.4.2.1.3.3

Add $49$ and $0$ .

${(49 - y^{2})}^{1} = {(- x)}^{2}$

Step 3.4.2.1.4

Simplify.

$49 - y^{2} = {(- x)}^{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(- x)}^{2}$ .

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Step 3.4.3.1.1

Apply the product rule to $- x$ .

$49 - y^{2} = {(- 1)}^{2} x^{2}$

Step 3.4.3.1.2

Raise $- 1$ to the power of $2$ .

$49 - y^{2} = 1 x^{2}$

Step 3.4.3.1.3

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

$49 - y^{2} = x^{2}$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Subtract $49$ from both sides of the equation.

$- y^{2} = x^{2} - 49$

Step 3.5.2

Divide each term in $- y^{2} = x^{2} - 49$ by $- 1$ and simplify.

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Step 3.5.2.1

Divide each term in $- y^{2} = x^{2} - 49$ by $- 1$ .

$\frac{- y^{2}}{- 1} = \frac{x^{2}}{- 1} + \frac{- 49}{- 1}$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} = \frac{x^{2}}{- 1} + \frac{- 49}{- 1}$

Step 3.5.2.2.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{x^{2}}{- 1} + \frac{- 49}{- 1}$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Simplify each term.

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Step 3.5.2.3.1.1

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

$y^{2} = - 1 \cdot x^{2} + \frac{- 49}{- 1}$

Step 3.5.2.3.1.2

Rewrite $- 1 \cdot x^{2}$ as $- x^{2}$ .

$y^{2} = - x^{2} + \frac{- 49}{- 1}$

Step 3.5.2.3.1.3

Divide $- 49$ by $- 1$ .

$y^{2} = - x^{2} + 49$

Step 3.5.3

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

$y = \pm \sqrt{- x^{2} + 49}$

Step 3.5.4

Simplify $\pm \sqrt{- x^{2} + 49}$ .

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Step 3.5.4.1

Simplify the expression.

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Step 3.5.4.1.1

Rewrite $49$ as $7^{2}$ .

$y = \pm \sqrt{- x^{2} + 7^{2}}$

Step 3.5.4.1.2

Reorder $- x^{2}$ and $7^{2}$ .

$y = \pm \sqrt{7^{2} - x^{2}}$

Step 3.5.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 7$ and $b = x$ .

$y = \pm \sqrt{(7 + x) (7 - x)}$

Step 3.5.5

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

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Step 3.5.5.1

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

$y = \sqrt{(7 + x) (7 - x)}$

Step 3.5.5.2

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