Algebra Examples

$\cos (2 θ) = \cos^{2} (θ) - \frac{1}{2}$

Step 1

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

$1 - 2 \sin^{2} (θ) = \cos^{2} (θ) - \frac{1}{2}$

Step 2

Subtract $1$ from both sides of the equation.

$- 2 \sin^{2} (θ) = \cos^{2} (θ) - \frac{1}{2} - 1$

Step 3

Subtract $\cos^{2} (θ)$ from both sides of the equation.

$- 2 \sin^{2} (θ) - \cos^{2} (θ) = - \frac{1}{2} - 1$

Step 4

Simplify the right side.

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Step 4.1

Simplify $- \frac{1}{2} - 1$ .

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Step 4.1.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 2 \sin^{2} (θ) - \cos^{2} (θ) = - \frac{1}{2} - 1 \cdot \frac{2}{2}$

Step 4.1.2

Combine $- 1$ and $\frac{2}{2}$ .

$- 2 \sin^{2} (θ) - \cos^{2} (θ) = - \frac{1}{2} + \frac{- 1 \cdot 2}{2}$

Step 4.1.3

Combine the numerators over the common denominator.

$- 2 \sin^{2} (θ) - \cos^{2} (θ) = \frac{- 1 - 1 \cdot 2}{2}$

Step 4.1.4

Simplify the numerator.

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Step 4.1.4.1

Multiply $- 1$ by $2$ .

$- 2 \sin^{2} (θ) - \cos^{2} (θ) = \frac{- 1 - 2}{2}$

Step 4.1.4.2

Subtract $2$ from $- 1$ .

$- 2 \sin^{2} (θ) - \cos^{2} (θ) = \frac{- 3}{2}$

Step 4.1.5

Move the negative in front of the fraction.

$- 2 \sin^{2} (θ) - \cos^{2} (θ) = - \frac{3}{2}$

Step 5

Solve the equation for $θ$ .

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Step 5.1

Add $\frac{3}{2}$ to both sides of the equation.

$- 2 \sin^{2} (θ) - \cos^{2} (θ) + \frac{3}{2} = 0$

Step 5.2

Replace $\cos^{2} (θ)$ with $1 - \sin^{2} (θ)$ .

$- 2 \sin^{2} (θ) - (1 - \sin^{2} (θ)) + \frac{3}{2} = 0$

Step 5.3

Solve for $θ$ .

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Step 5.3.1

Simplify $- 2 \sin^{2} (θ) - (1 - \sin^{2} (θ)) + \frac{3}{2}$ .

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Step 5.3.1.1

Find the common denominator.

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Step 5.3.1.1.1

Write $- 2 \sin^{2} (θ)$ as a fraction with denominator $1$ .

$\frac{- 2 \sin^{2} (θ)}{1} - (1 - \sin^{2} (θ)) + \frac{3}{2} = 0$

Step 5.3.1.1.2

Multiply $\frac{- 2 \sin^{2} (θ)}{1}$ by $\frac{2}{2}$ .

$\frac{- 2 \sin^{2} (θ)}{1} \cdot \frac{2}{2} - (1 - \sin^{2} (θ)) + \frac{3}{2} = 0$

Step 5.3.1.1.3

Multiply $\frac{- 2 \sin^{2} (θ)}{1}$ by $\frac{2}{2}$ .

$\frac{- 2 \sin^{2} (θ) \cdot 2}{2} - (1 - \sin^{2} (θ)) + \frac{3}{2} = 0$

Step 5.3.1.1.4

Write $- (1 - \sin^{2} (θ))$ as a fraction with denominator $1$ .

$\frac{- 2 \sin^{2} (θ) \cdot 2}{2} + \frac{- (1 - \sin^{2} (θ))}{1} + \frac{3}{2} = 0$

Step 5.3.1.1.5

Multiply $\frac{- (1 - \sin^{2} (θ))}{1}$ by $\frac{2}{2}$ .

$\frac{- 2 \sin^{2} (θ) \cdot 2}{2} + \frac{- (1 - \sin^{2} (θ))}{1} \cdot \frac{2}{2} + \frac{3}{2} = 0$

Step 5.3.1.1.6

Multiply $\frac{- (1 - \sin^{2} (θ))}{1}$ by $\frac{2}{2}$ .

$\frac{- 2 \sin^{2} (θ) \cdot 2}{2} + \frac{- (1 - \sin^{2} (θ)) \cdot 2}{2} + \frac{3}{2} = 0$

Step 5.3.1.2

Combine the numerators over the common denominator.

$\frac{- 2 \sin^{2} (θ) \cdot 2 - (1 - \sin^{2} (θ)) \cdot 2 + 3}{2} = 0$

Step 5.3.1.3

Simplify each term.

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Step 5.3.1.3.1

Multiply $2$ by $- 2$ .

$\frac{- 4 \sin^{2} (θ) - (1 - \sin^{2} (θ)) \cdot 2 + 3}{2} = 0$

Step 5.3.1.3.2

Apply the distributive property.

$\frac{- 4 \sin^{2} (θ) + (- 1 \cdot 1 - - \sin^{2} (θ)) \cdot 2 + 3}{2} = 0$

Step 5.3.1.3.3

Multiply $- 1$ by $1$ .

$\frac{- 4 \sin^{2} (θ) + (- 1 - - \sin^{2} (θ)) \cdot 2 + 3}{2} = 0$

Step 5.3.1.3.4

Multiply $- - \sin^{2} (θ)$ .

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Step 5.3.1.3.4.1

Multiply $- 1$ by $- 1$ .

$\frac{- 4 \sin^{2} (θ) + (- 1 + 1 \sin^{2} (θ)) \cdot 2 + 3}{2} = 0$

Step 5.3.1.3.4.2

Multiply $\sin^{2} (θ)$ by $1$ .

$\frac{- 4 \sin^{2} (θ) + (- 1 + \sin^{2} (θ)) \cdot 2 + 3}{2} = 0$

Step 5.3.1.3.5

Apply the distributive property.

$\frac{- 4 \sin^{2} (θ) - 1 \cdot 2 + \sin^{2} (θ) \cdot 2 + 3}{2} = 0$

Step 5.3.1.3.6

Multiply $- 1$ by $2$ .

$\frac{- 4 \sin^{2} (θ) - 2 + \sin^{2} (θ) \cdot 2 + 3}{2} = 0$

Step 5.3.1.3.7

Move $2$ to the left of $\sin^{2} (θ)$ .

$\frac{- 4 \sin^{2} (θ) - 2 + 2 \sin^{2} (θ) + 3}{2} = 0$

Step 5.3.1.4

Simplify terms.

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Step 5.3.1.4.1

Add $- 4 \sin^{2} (θ)$ and $2 \sin^{2} (θ)$ .

$\frac{- 2 \sin^{2} (θ) - 2 + 3}{2} = 0$

Step 5.3.1.4.2

Add $- 2$ and $3$ .

$\frac{- 2 \sin^{2} (θ) + 1}{2} = 0$

Step 5.3.1.4.3

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

$\frac{- (2 \sin^{2} (θ)) + 1}{2} = 0$

Step 5.3.1.4.4

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

$\frac{- (2 \sin^{2} (θ)) - 1 (- 1)}{2} = 0$

Step 5.3.1.4.5

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

$\frac{- (2 \sin^{2} (θ) - 1)}{2} = 0$

Step 5.3.1.4.6

Simplify the expression.

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Step 5.3.1.4.6.1

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

$\frac{- 1 (2 \sin^{2} (θ) - 1)}{2} = 0$

Step 5.3.1.4.6.2

Move the negative in front of the fraction.

$- \frac{2 \sin^{2} (θ) - 1}{2} = 0$

Step 5.3.2

Set the numerator equal to zero.

$2 \sin^{2} (θ) - 1 = 0$

Step 5.3.3

Solve the equation for $\sin (θ)$ .

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Step 5.3.3.1

Add $1$ to both sides of the equation.

$2 \sin^{2} (θ) = 1$

Step 5.3.3.2

Divide each term in $2 \sin^{2} (θ) = 1$ by $2$ and simplify.

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Step 5.3.3.2.1

Divide each term in $2 \sin^{2} (θ) = 1$ by $2$ .

$\frac{2 \sin^{2} (θ)}{2} = \frac{1}{2}$

Step 5.3.3.2.2

Simplify the left side.

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Step 5.3.3.2.2.1

Cancel the common factor of $2$ .

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Step 5.3.3.2.2.1.1

Cancel the common factor.

$\frac{2 \sin^{2} (θ)}{2} = \frac{1}{2}$

Step 5.3.3.2.2.1.2

Divide $\sin^{2} (θ)$ by $1$ .

$\sin^{2} (θ) = \frac{1}{2}$

Step 5.3.3.3

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

$\sin (θ) = \pm \sqrt{\frac{1}{2}}$

Step 5.3.3.4

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 5.3.3.4.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\sin (θ) = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 5.3.3.4.2

Any root of $1$ is $1$ .

$\sin (θ) = \pm \frac{1}{\sqrt{2}}$

Step 5.3.3.4.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (θ) = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5.3.3.4.4

Combine and simplify the denominator.

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Step 5.3.3.4.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (θ) = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.3.3.4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (θ) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 5.3.3.4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (θ) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 5.3.3.4.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (θ) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5.3.3.4.4.5

Add $1$ and $1$ .

$\sin (θ) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 5.3.3.4.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 5.3.3.4.4.6.1

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

$\sin (θ) = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 5.3.3.4.4.6.2

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

$\sin (θ) = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 5.3.3.4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\sin (θ) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.3.3.4.4.6.4

Cancel the common factor of $2$ .

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Step 5.3.3.4.4.6.4.1

Cancel the common factor.

$\sin (θ) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.3.3.4.4.6.4.2

Rewrite the expression.

$\sin (θ) = \pm \frac{\sqrt{2}}{2^{1}}$

Step 5.3.3.4.4.6.5

Evaluate the exponent.

$\sin (θ) = \pm \frac{\sqrt{2}}{2}$

Step 5.3.3.5

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

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Step 5.3.3.5.1

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

$\sin (θ) = \frac{\sqrt{2}}{2}$

Step 5.3.3.5.2

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

$\sin (θ) = - \frac{\sqrt{2}}{2}$

Step 5.3.3.5.3

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

$\sin (θ) = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 5.3.4

Set up each of the solutions to solve for $θ$ .

$\sin (θ) = \frac{\sqrt{2}}{2}$

$\sin (θ) = - \frac{\sqrt{2}}{2}$

Step 5.3.5

Solve for $θ$ in $\sin (θ) = \frac{\sqrt{2}}{2}$ .

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Step 5.3.5.1

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

$θ = \arcsin (\frac{\sqrt{2}}{2})$

Step 5.3.5.2

Simplify the right side.

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Step 5.3.5.2.1

The exact value of $\arcsin (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$θ = \frac{π}{4}$

Step 5.3.5.3

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.

$θ = π - \frac{π}{4}$

Step 5.3.5.4

Simplify $π - \frac{π}{4}$ .

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Step 5.3.5.4.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$θ = π \cdot \frac{4}{4} - \frac{π}{4}$

Step 5.3.5.4.2

Combine fractions.

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Step 5.3.5.4.2.1

Combine $π$ and $\frac{4}{4}$ .

$θ = \frac{π \cdot 4}{4} - \frac{π}{4}$

Step 5.3.5.4.2.2

Combine the numerators over the common denominator.

$θ = \frac{π \cdot 4 - π}{4}$

Step 5.3.5.4.3

Simplify the numerator.

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Step 5.3.5.4.3.1

Move $4$ to the left of $π$ .

$θ = \frac{4 \cdot π - π}{4}$

Step 5.3.5.4.3.2

Subtract $π$ from $4 π$ .

$θ = \frac{3 π}{4}$

Step 5.3.5.5

Find the period of $\sin (θ)$ .

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Step 5.3.5.5.1

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

$\frac{2 π}{| b |}$

Step 5.3.5.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.3.5.5.3

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

$\frac{2 π}{1}$

Step 5.3.5.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.3.5.6

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

$θ = \frac{π}{4} + 2 π n, \frac{3 π}{4} + 2 π n$ , for any integer $n$

Step 5.3.6

Solve for $θ$ in $\sin (θ) = - \frac{\sqrt{2}}{2}$ .

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Step 5.3.6.1

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

$θ = \arcsin (- \frac{\sqrt{2}}{2})$

Step 5.3.6.2

Simplify the right side.

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Step 5.3.6.2.1

The exact value of $\arcsin (- \frac{\sqrt{2}}{2})$ is $- \frac{π}{4}$ .

$θ = - \frac{π}{4}$

Step 5.3.6.3

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.

$θ = 2 π + \frac{π}{4} + π$

Step 5.3.6.4

Simplify the expression to find the second solution.

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Step 5.3.6.4.1

Subtract $2 π$ from $2 π + \frac{π}{4} + π$ .

$θ = 2 π + \frac{π}{4} + π - 2 π$

Step 5.3.6.4.2

The resulting angle of $\frac{5 π}{4}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{4} + π$ .

$θ = \frac{5 π}{4}$

Step 5.3.6.5

Find the period of $\sin (θ)$ .

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Step 5.3.6.5.1

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

$\frac{2 π}{| b |}$

Step 5.3.6.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.3.6.5.3

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

$\frac{2 π}{1}$

Step 5.3.6.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.3.6.6

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

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Step 5.3.6.6.1

Add $2 π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + 2 π$

Step 5.3.6.6.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 5.3.6.6.3

Combine fractions.

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Step 5.3.6.6.3.1

Combine $2 π$ and $\frac{4}{4}$ .

$\frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 5.3.6.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 4 - π}{4}$

Step 5.3.6.6.4

Simplify the numerator.

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Step 5.3.6.6.4.1

Multiply $4$ by $2$ .

$\frac{8 π - π}{4}$

Step 5.3.6.6.4.2

Subtract $π$ from $8 π$ .

$\frac{7 π}{4}$

Step 5.3.6.6.5

List the new angles.

$θ = \frac{7 π}{4}$

Step 5.3.6.7

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

$θ = \frac{5 π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 5.3.7

List all of the solutions.

$θ = \frac{π}{4} + 2 π n, \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 5.3.8

Consolidate the answers.

$θ = \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$