Finite Math Examples

$79 \sin^{2} (x) = \cos (2 x)$

Step 1

Subtract $\cos (2 x)$ from both sides of the equation.

$79 \sin^{2} (x) - \cos (2 x) = 0$

Step 2

Simplify the left side of the equation.

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Step 2.1

Simplify each term.

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Step 2.1.1

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

$79 \sin^{2} (x) - (1 - 2 \sin^{2} (x)) = 0$

Step 2.1.2

Apply the distributive property.

$79 \sin^{2} (x) - 1 \cdot 1 - (- 2 \sin^{2} (x)) = 0$

Step 2.1.3

Multiply $- 1$ by $1$ .

$79 \sin^{2} (x) - 1 - (- 2 \sin^{2} (x)) = 0$

Step 2.1.4

Multiply $- 2$ by $- 1$ .

$79 \sin^{2} (x) - 1 + 2 \sin^{2} (x) = 0$

Step 2.2

Add $79 \sin^{2} (x)$ and $2 \sin^{2} (x)$ .

$- 1 + 81 \sin^{2} (x) = 0$

Step 3

Factor $- 1 + 81 \sin^{2} (x)$ .

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Step 3.1

Rewrite $81 \sin^{2} (x)$ as ${(9 \sin (x))}^{2}$ .

$- 1 + {(9 \sin (x))}^{2} = 0$

Step 3.2

Rewrite $1$ as $1^{2}$ .

$- 1^{2} + {(9 \sin (x))}^{2} = 0$

Step 3.3

Reorder $- 1^{2}$ and ${(9 \sin (x))}^{2}$ .

${(9 \sin (x))}^{2} - 1^{2} = 0$

Step 3.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 9 \sin (x)$ and $b = 1$ .

$(9 \sin (x) + 1) (9 \sin (x) - 1) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$9 \sin (x) + 1 = 0$

$9 \sin (x) - 1 = 0$

Step 5

Set $9 \sin (x) + 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $9 \sin (x) + 1$ equal to $0$ .

$9 \sin (x) + 1 = 0$

Step 5.2

Solve $9 \sin (x) + 1 = 0$ for $x$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$9 \sin (x) = - 1$

Step 5.2.2

Divide each term in $9 \sin (x) = - 1$ by $9$ and simplify.

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Step 5.2.2.1

Divide each term in $9 \sin (x) = - 1$ by $9$ .

$\frac{9 \sin (x)}{9} = \frac{- 1}{9}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $9$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{9 \sin (x)}{9} = \frac{- 1}{9}$

Step 5.2.2.2.1.2

Divide $\sin (x)$ by $1$ .

$\sin (x) = \frac{- 1}{9}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

$\sin (x) = - \frac{1}{9}$

Step 5.2.3

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (- \frac{1}{9})$

Step 5.2.4

Simplify the right side.

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Step 5.2.4.1

Evaluate $\arcsin (- \frac{1}{9})$ .

$x = - 0.11134101$

Step 5.2.5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) + 0.11134101 + 3.14159265$

Step 5.2.6

Simplify the expression to find the second solution.

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Step 5.2.6.1

Subtract $2 π$ from $2 (3.14159265) + 0.11134101 + 3.14159265$ .

$x = 2 (3.14159265) + 0.11134101 + 3.14159265 - 2 π$

Step 5.2.6.2

The resulting angle of $3.25293366$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.11134101 + 3.14159265$ .

$x = 3.25293366$

Step 5.2.7

Find the period of $\sin (x)$ .

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Step 5.2.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 5.2.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 5.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 5.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.8

Add $2 π$ to every negative angle to get positive angles.

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Step 5.2.8.1

Add $2 π$ to $- 0.11134101$ to find the positive angle.

$- 0.11134101 + 2 π$

Step 5.2.8.2

Subtract $0.11134101$ from $2 π$ .

$6.17184429$

Step 5.2.8.3

List the new angles.

$x = 6.17184429$

Step 5.2.9

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 3.25293366 + 2 π n, 6.17184429 + 2 π n$ , for any integer $n$

Step 6

Set $9 \sin (x) - 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $9 \sin (x) - 1$ equal to $0$ .

$9 \sin (x) - 1 = 0$

Step 6.2

Solve $9 \sin (x) - 1 = 0$ for $x$ .

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Step 6.2.1

Add $1$ to both sides of the equation.

$9 \sin (x) = 1$

Step 6.2.2

Divide each term in $9 \sin (x) = 1$ by $9$ and simplify.

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Step 6.2.2.1

Divide each term in $9 \sin (x) = 1$ by $9$ .

$\frac{9 \sin (x)}{9} = \frac{1}{9}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $9$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{9 \sin (x)}{9} = \frac{1}{9}$

Step 6.2.2.2.1.2

Divide $\sin (x)$ by $1$ .

$\sin (x) = \frac{1}{9}$

Step 6.2.3

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (\frac{1}{9})$

Step 6.2.4

Simplify the right side.

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Step 6.2.4.1

Evaluate $\arcsin (\frac{1}{9})$ .

$x = 0.11134101$

Step 6.2.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = (3.14159265) - 0.11134101$

Step 6.2.6

Solve for $x$ .

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Step 6.2.6.1

Remove parentheses.

$x = 3.14159265 - 0.11134101$

Step 6.2.6.2

Remove parentheses.

$x = (3.14159265) - 0.11134101$

Step 6.2.6.3

Subtract $0.11134101$ from $3.14159265$ .

$x = 3.03025163$

Step 6.2.7

Find the period of $\sin (x)$ .

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Step 6.2.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 6.2.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 6.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 6.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.2.8

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 0.11134101 + 2 π n, 3.03025163 + 2 π n$ , for any integer $n$

Step 7

The final solution is all the values that make $(9 \sin (x) + 1) (9 \sin (x) - 1) = 0$ true.

$x = 3.25293366 + 2 π n, 6.17184429 + 2 π n, 0.11134101 + 2 π n, 3.03025163 + 2 π n$ , for any integer $n$

Step 8

Consolidate the answers.

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Step 8.1

Consolidate $3.25293366 + 2 π n$ and $0.11134101 + 2 π n$ to $0.11134101 + π n$ .

$x = 0.11134101 + π n, 6.17184429 + 2 π n, 3.03025163 + 2 π n$ , for any integer $n$

Step 8.2

Consolidate $6.17184429 + 2 π n$ and $3.03025163 + 2 π n$ to $3.03025163 + π n$ .

$x = 0.11134101 + π n, 3.03025163 + π n$ , for any integer $n$