Trigonometry Examples

$\arcsin (\sqrt{1 - \frac{2}{y + 1}}) = x$

Step 1

Take the inverse arcsine of both sides of the equation to extract $y$ from inside the arcsine.

$\sqrt{1 - \frac{2}{y + 1}} = \sin (x)$

Step 2

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

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

Step 3

Simplify each side of the equation.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 4

Solve for $y$ .

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Step 4.1

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

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Step 4.1.1

Subtract $1$ from both sides of the equation.

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

Step 4.1.2

Reorder $\sin^{2} (x)$ and $- 1$ .

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

Step 4.1.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 4.1.4

Factor $- 1$ out of $\sin^{2} (x)$ .

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

Step 4.1.5

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (x))$ .

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

Step 4.1.6

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

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

Step 4.1.7

Apply pythagorean identity.

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

Step 4.2

Find the LCD of the terms in the equation.

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Step 4.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y + 1, 1$

Step 4.2.2

Remove parentheses.

$y + 1, 1$

Step 4.2.3

The LCM of one and any expression is the expression.

$y + 1$

Step 4.3

Multiply each term in $- \frac{2}{y + 1} = - \cos^{2} (x)$ by $y + 1$ to eliminate the fractions.

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Step 4.3.1

Multiply each term in $- \frac{2}{y + 1} = - \cos^{2} (x)$ by $y + 1$ .

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $y + 1$ .

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Step 4.3.2.1.1

Move the leading negative in $- \frac{2}{y + 1}$ into the numerator.

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

Step 4.3.2.1.2

Cancel the common factor.

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

Step 4.3.2.1.3

Rewrite the expression.

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

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Apply the distributive property.

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

Step 4.3.3.2

Simplify the expression.

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Step 4.3.3.2.1

Multiply $- 1$ by $1$ .

$- 2 = - \cos^{2} (x) y - \cos^{2} (x)$

Step 4.3.3.2.2

Reorder factors in $- \cos^{2} (x) y - \cos^{2} (x)$ .

$- 2 = - y \cos^{2} (x) - \cos^{2} (x)$

Step 4.4

Solve the equation.

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Step 4.4.1

Rewrite the equation as $- y \cos^{2} (x) - \cos^{2} (x) = - 2$ .

$- y \cos^{2} (x) - \cos^{2} (x) = - 2$

Step 4.4.2

Add $\cos^{2} (x)$ to both sides of the equation.

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

Step 4.4.3

Divide each term in $- y \cos^{2} (x) = - 2 + \cos^{2} (x)$ by $- \cos^{2} (x)$ and simplify.

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Step 4.4.3.1

Divide each term in $- y \cos^{2} (x) = - 2 + \cos^{2} (x)$ by $- \cos^{2} (x)$ .

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

Step 4.4.3.2

Simplify the left side.

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Step 4.4.3.2.1

Dividing two negative values results in a positive value.

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

Step 4.4.3.2.2

Cancel the common factor of $\cos^{2} (x)$ .

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Step 4.4.3.2.2.1

Cancel the common factor.

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

Step 4.4.3.2.2.2

Divide $y$ by $1$ .

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

Step 4.4.3.3

Simplify the right side.

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Step 4.4.3.3.1

Simplify each term.

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Step 4.4.3.3.1.1

Multiply by $1$ .

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

Step 4.4.3.3.1.2

Separate fractions.

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

Step 4.4.3.3.1.3

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

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

Step 4.4.3.3.1.4

Multiply $- 1$ by $1$ .

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

Step 4.4.3.3.1.5

Divide $- 2$ by $- 1$ .

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

Step 4.4.3.3.1.6

Cancel the common factor of $\cos^{2} (x)$ .

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Step 4.4.3.3.1.6.1

Cancel the common factor.

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

Step 4.4.3.3.1.6.2

Rewrite the expression.

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

Step 4.4.3.3.1.6.3

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

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

Step 4.4.3.3.1.7

Multiply $- 1$ by $1$ .

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