Trigonometry Examples

$- \frac{1}{7} \cdot \sqrt{16 - x^{2}}$

Step 1

Interchange the variables.

$x = - \frac{1}{7} \cdot \sqrt{16 - y^{2}}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- \frac{1}{7} \cdot \sqrt{16 - y^{2}} = x$ .

$- \frac{1}{7} \cdot \sqrt{16 - y^{2}} = x$

Step 2.2

Multiply both sides of the equation by $- 7$ .

$- 7 (- \frac{1}{7} \cdot \sqrt{16 - y^{2}}) = - 7 x$

Step 2.3

Simplify the left side.

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Step 2.3.1

Simplify $- 7 (- \frac{1}{7} \cdot \sqrt{16 - y^{2}})$ .

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Step 2.3.1.1

Rewrite $16$ as $4^{2}$ .

$- 7 (- \frac{1}{7} \cdot \sqrt{4^{2} - y^{2}}) = - 7 x$

Step 2.3.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = y$ .

$- 7 (- \frac{1}{7} \cdot \sqrt{(4 + y) (4 - y)}) = - 7 x$

Step 2.3.1.3

Simplify terms.

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Step 2.3.1.3.1

Combine $\sqrt{(4 + y) (4 - y)}$ and $\frac{1}{7}$ .

$- 7 (- \frac{\sqrt{(4 + y) (4 - y)}}{7}) = - 7 x$

Step 2.3.1.3.2

Cancel the common factor of $7$ .

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Step 2.3.1.3.2.1

Move the leading negative in $- \frac{\sqrt{(4 + y) (4 - y)}}{7}$ into the numerator.

$- 7 \frac{- \sqrt{(4 + y) (4 - y)}}{7} = - 7 x$

Step 2.3.1.3.2.2

Factor $7$ out of $- 7$ .

$7 (- 1) \frac{- \sqrt{(4 + y) (4 - y)}}{7} = - 7 x$

Step 2.3.1.3.2.3

Cancel the common factor.

$7 \cdot - 1 \frac{- \sqrt{(4 + y) (4 - y)}}{7} = - 7 x$

Step 2.3.1.3.2.4

Rewrite the expression.

$- - \sqrt{(4 + y) (4 - y)} = - 7 x$

Step 2.3.1.3.3

Multiply.

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Step 2.3.1.3.3.1

Multiply $- 1$ by $- 1$ .

$1 \sqrt{(4 + y) (4 - y)} = - 7 x$

Step 2.3.1.3.3.2

Multiply $\sqrt{(4 + y) (4 - y)}$ by $1$ .

$\sqrt{(4 + y) (4 - y)} = - 7 x$

Step 2.4

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

${\sqrt{(4 + y) (4 - y)}}^{2} = {(- 7 x)}^{2}$

Step 2.5

Simplify each side of the equation.

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(4 + y) (4 - y)}$ as ${((4 + y) (4 - y))}^{\frac{1}{2}}$ .

${({((4 + y) (4 - y))}^{\frac{1}{2}})}^{2} = {(- 7 x)}^{2}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Simplify ${({((4 + y) (4 - y))}^{\frac{1}{2}})}^{2}$ .

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Step 2.5.2.1.1

Multiply the exponents in ${({((4 + y) (4 - y))}^{\frac{1}{2}})}^{2}$ .

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Step 2.5.2.1.1.1

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

${((4 + y) (4 - y))}^{\frac{1}{2} \cdot 2} = {(- 7 x)}^{2}$

Step 2.5.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.5.2.1.1.2.1

Cancel the common factor.

${((4 + y) (4 - y))}^{\frac{1}{2} \cdot 2} = {(- 7 x)}^{2}$

Step 2.5.2.1.1.2.2

Rewrite the expression.

${((4 + y) (4 - y))}^{1} = {(- 7 x)}^{2}$

Step 2.5.2.1.2

Expand $(4 + y) (4 - y)$ using the FOIL Method.

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Step 2.5.2.1.2.1

Apply the distributive property.

${(4 (4 - y) + y (4 - y))}^{1} = {(- 7 x)}^{2}$

Step 2.5.2.1.2.2

Apply the distributive property.

${(4 \cdot 4 + 4 (- y) + y (4 - y))}^{1} = {(- 7 x)}^{2}$

Step 2.5.2.1.2.3

Apply the distributive property.

${(4 \cdot 4 + 4 (- y) + y \cdot 4 + y (- y))}^{1} = {(- 7 x)}^{2}$

Step 2.5.2.1.3

Simplify and combine like terms.

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Step 2.5.2.1.3.1

Simplify each term.

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Step 2.5.2.1.3.1.1

Multiply $4$ by $4$ .

${(16 + 4 (- y) + y \cdot 4 + y (- y))}^{1} = {(- 7 x)}^{2}$

Step 2.5.2.1.3.1.2

Multiply $- 1$ by $4$ .

${(16 - 4 y + y \cdot 4 + y (- y))}^{1} = {(- 7 x)}^{2}$

Step 2.5.2.1.3.1.3

Move $4$ to the left of $y$ .

${(16 - 4 y + 4 \cdot y + y (- y))}^{1} = {(- 7 x)}^{2}$

Step 2.5.2.1.3.1.4

Rewrite using the commutative property of multiplication.

${(16 - 4 y + 4 y - y \cdot y)}^{1} = {(- 7 x)}^{2}$

Step 2.5.2.1.3.1.5

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

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Step 2.5.2.1.3.1.5.1

Move $y$ .

${(16 - 4 y + 4 y - (y \cdot y))}^{1} = {(- 7 x)}^{2}$

Step 2.5.2.1.3.1.5.2

Multiply $y$ by $y$ .

${(16 - 4 y + 4 y - y^{2})}^{1} = {(- 7 x)}^{2}$

Step 2.5.2.1.3.2

Add $- 4 y$ and $4 y$ .

${(16 + 0 - y^{2})}^{1} = {(- 7 x)}^{2}$

Step 2.5.2.1.3.3

Add $16$ and $0$ .

${(16 - y^{2})}^{1} = {(- 7 x)}^{2}$

Step 2.5.2.1.4

Simplify.

$16 - y^{2} = {(- 7 x)}^{2}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Simplify ${(- 7 x)}^{2}$ .

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Step 2.5.3.1.1

Apply the product rule to $- 7 x$ .

$16 - y^{2} = {(- 7)}^{2} x^{2}$

Step 2.5.3.1.2

Raise $- 7$ to the power of $2$ .

$16 - y^{2} = 49 x^{2}$

Step 2.6

Solve for $y$ .

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Step 2.6.1

Subtract $16$ from both sides of the equation.

$- y^{2} = 49 x^{2} - 16$

Step 2.6.2

Divide each term in $- y^{2} = 49 x^{2} - 16$ by $- 1$ and simplify.

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Step 2.6.2.1

Divide each term in $- y^{2} = 49 x^{2} - 16$ by $- 1$ .

$\frac{- y^{2}}{- 1} = \frac{49 x^{2}}{- 1} + \frac{- 16}{- 1}$

Step 2.6.2.2

Simplify the left side.

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Step 2.6.2.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} = \frac{49 x^{2}}{- 1} + \frac{- 16}{- 1}$

Step 2.6.2.2.2

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

$y^{2} = \frac{49 x^{2}}{- 1} + \frac{- 16}{- 1}$

Step 2.6.2.3

Simplify the right side.

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Step 2.6.2.3.1

Simplify each term.

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Step 2.6.2.3.1.1

Move the negative one from the denominator of $\frac{49 x^{2}}{- 1}$ .

$y^{2} = - 1 \cdot (49 x^{2}) + \frac{- 16}{- 1}$

Step 2.6.2.3.1.2

Rewrite $- 1 \cdot (49 x^{2})$ as $- (49 x^{2})$ .

$y^{2} = - (49 x^{2}) + \frac{- 16}{- 1}$

Step 2.6.2.3.1.3

Multiply $49$ by $- 1$ .

$y^{2} = - 49 x^{2} + \frac{- 16}{- 1}$

Step 2.6.2.3.1.4

Divide $- 16$ by $- 1$ .

$y^{2} = - 49 x^{2} + 16$

Step 2.6.3

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

$y = \pm \sqrt{- 49 x^{2} + 16}$

Step 2.6.4

Simplify $\pm \sqrt{- 49 x^{2} + 16}$ .

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Step 2.6.4.1

Simplify the expression.

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Step 2.6.4.1.1

Rewrite $16$ as $4^{2}$ .

$y = \pm \sqrt{- 49 x^{2} + 4^{2}}$

Step 2.6.4.1.2

Rewrite $49 x^{2}$ as ${(7 x)}^{2}$ .

$y = \pm \sqrt{- {(7 x)}^{2} + 4^{2}}$

Step 2.6.4.1.3

Reorder $- {(7 x)}^{2}$ and $4^{2}$ .

$y = \pm \sqrt{4^{2} - {(7 x)}^{2}}$

Step 2.6.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = 7 x$ .

$y = \pm \sqrt{(4 + 7 x) (4 - (7 x))}$

Step 2.6.4.3

Multiply $7$ by $- 1$ .

$y = \pm \sqrt{(4 + 7 x) (4 - 7 x)}$

Step 2.6.5

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

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Step 2.6.5.1

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