Trigonometry Examples

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

Step 1

Subtract $\frac{\sqrt{3}}{2}$ from both sides of the equation.

$\sin^{2} (x) - \cos^{2} (x) - \frac{\sqrt{3}}{2} = 0$

Step 2

Replace the $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$(1 - \cos^{2} (x)) - \cos^{2} (x) - \frac{\sqrt{3}}{2} = 0$

Step 3

Subtract $\cos^{2} (x)$ from $- \cos^{2} (x)$ .

$1 - 2 \cos^{2} (x) - \frac{\sqrt{3}}{2} = 0$

Step 4

Reorder the polynomial.

$- 2 \cos^{2} (x) + 1 - \frac{\sqrt{3}}{2} = 0$

Step 5

Move all terms not containing $\cos (x)$ to the right side of the equation.

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Step 5.1

Subtract $1$ from both sides of the equation.

$- 2 \cos^{2} (x) - \frac{\sqrt{3}}{2} = - 1$

Step 5.2

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

$- 2 \cos^{2} (x) = - 1 + \frac{\sqrt{3}}{2}$

Step 6

Divide each term in $- 2 \cos^{2} (x) = - 1 + \frac{\sqrt{3}}{2}$ by $- 2$ and simplify.

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Step 6.1

Divide each term in $- 2 \cos^{2} (x) = - 1 + \frac{\sqrt{3}}{2}$ by $- 2$ .

$\frac{- 2 \cos^{2} (x)}{- 2} = \frac{- 1}{- 2} + \frac{\frac{\sqrt{3}}{2}}{- 2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 2 \cos^{2} (x)}{- 2} = \frac{- 1}{- 2} + \frac{\frac{\sqrt{3}}{2}}{- 2}$

Step 6.2.1.2

Divide $\cos^{2} (x)$ by $1$ .

$\cos^{2} (x) = \frac{- 1}{- 2} + \frac{\frac{\sqrt{3}}{2}}{- 2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Dividing two negative values results in a positive value.

$\cos^{2} (x) = \frac{1}{2} + \frac{\frac{\sqrt{3}}{2}}{- 2}$

Step 6.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$\cos^{2} (x) = \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \frac{1}{- 2}$

Step 6.3.1.3

Move the negative in front of the fraction.

$\cos^{2} (x) = \frac{1}{2} + \frac{\sqrt{3}}{2} (- \frac{1}{2})$

Step 6.3.1.4

Multiply $\frac{\sqrt{3}}{2} (- \frac{1}{2})$ .

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Step 6.3.1.4.1

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{1}{2}$ .

$\cos^{2} (x) = \frac{1}{2} - \frac{\sqrt{3}}{2 \cdot 2}$

Step 6.3.1.4.2

Multiply $2$ by $2$ .

$\cos^{2} (x) = \frac{1}{2} - \frac{\sqrt{3}}{4}$

Step 7

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

$\cos (x) = \pm \sqrt{\frac{1}{2} - \frac{\sqrt{3}}{4}}$

Step 8

Simplify $\pm \sqrt{\frac{1}{2} - \frac{\sqrt{3}}{4}}$ .

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Step 8.1

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

$\cos (x) = \pm \sqrt{\frac{1}{2} \cdot \frac{2}{2} - \frac{\sqrt{3}}{4}}$

Step 8.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.1

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

$\cos (x) = \pm \sqrt{\frac{2}{2 \cdot 2} - \frac{\sqrt{3}}{4}}$

Step 8.2.2

Multiply $2$ by $2$ .

$\cos (x) = \pm \sqrt{\frac{2}{4} - \frac{\sqrt{3}}{4}}$

Step 8.3

Combine the numerators over the common denominator.

$\cos (x) = \pm \sqrt{\frac{2 - \sqrt{3}}{4}}$

Step 8.4

Rewrite $\sqrt{\frac{2 - \sqrt{3}}{4}}$ as $\frac{\sqrt{2 - \sqrt{3}}}{\sqrt{4}}$ .

$\cos (x) = \pm \frac{\sqrt{2 - \sqrt{3}}}{\sqrt{4}}$

Step 8.5

Simplify the denominator.

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Step 8.5.1

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

$\cos (x) = \pm \frac{\sqrt{2 - \sqrt{3}}}{\sqrt{2^{2}}}$

Step 8.5.2

Pull terms out from under the radical, assuming positive real numbers.

$\cos (x) = \pm \frac{\sqrt{2 - \sqrt{3}}}{2}$

Step 9

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

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Step 9.1

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

$\cos (x) = \frac{\sqrt{2 - \sqrt{3}}}{2}$

Step 9.2

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

$\cos (x) = - \frac{\sqrt{2 - \sqrt{3}}}{2}$

Step 9.3

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

$\cos (x) = \frac{\sqrt{2 - \sqrt{3}}}{2}, - \frac{\sqrt{2 - \sqrt{3}}}{2}$

Step 10

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

$\cos (x) = \frac{\sqrt{2 - \sqrt{3}}}{2}$

$\cos (x) = - \frac{\sqrt{2 - \sqrt{3}}}{2}$

Step 11

Solve for $x$ in $\cos (x) = \frac{\sqrt{2 - \sqrt{3}}}{2}$ .

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Step 11.1

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

$x = \arccos (\frac{\sqrt{2 - \sqrt{3}}}{2})$

Step 11.2

Simplify the right side.

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Step 11.2.1

Evaluate $\arccos (\frac{\sqrt{2 - \sqrt{3}}}{2})$ .

$x = \frac{5 π}{12}$

Step 11.3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{5 π}{12}$

Step 11.4

Simplify $2 π - \frac{5 π}{12}$ .

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Step 11.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$x = 2 π \cdot \frac{12}{12} - \frac{5 π}{12}$

Step 11.4.2

Combine fractions.

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Step 11.4.2.1

Combine $2 π$ and $\frac{12}{12}$ .

$x = \frac{2 π \cdot 12}{12} - \frac{5 π}{12}$

Step 11.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 12 - 5 π}{12}$

Step 11.4.3

Simplify the numerator.

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Step 11.4.3.1

Multiply $12$ by $2$ .

$x = \frac{24 π - 5 π}{12}$

Step 11.4.3.2

Subtract $5 π$ from $24 π$ .

$x = \frac{19 π}{12}$

Step 11.5

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

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Step 11.5.1

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

$\frac{2 π}{| b |}$

Step 11.5.2

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

$\frac{2 π}{| 1 |}$

Step 11.5.3

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

$\frac{2 π}{1}$

Step 11.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 11.6

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

$x = \frac{5 π}{12} + 2 π n, \frac{19 π}{12} + 2 π n$ , for any integer $n$

Step 12

Solve for $x$ in $\cos (x) = - \frac{\sqrt{2 - \sqrt{3}}}{2}$ .

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Step 12.1

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

$x = \arccos (- \frac{\sqrt{2 - \sqrt{3}}}{2})$

Step 12.2

Simplify the right side.

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Step 12.2.1

Evaluate $\arccos (- \frac{\sqrt{2 - \sqrt{3}}}{2})$ .

$x = \frac{7 π}{12}$

Step 12.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{7 π}{12}$

Step 12.4

Simplify $2 π - \frac{7 π}{12}$ .

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Step 12.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$x = 2 π \cdot \frac{12}{12} - \frac{7 π}{12}$

Step 12.4.2

Combine fractions.

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Step 12.4.2.1

Combine $2 π$ and $\frac{12}{12}$ .

$x = \frac{2 π \cdot 12}{12} - \frac{7 π}{12}$

Step 12.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 12 - 7 π}{12}$

Step 12.4.3

Simplify the numerator.

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Step 12.4.3.1

Multiply $12$ by $2$ .

$x = \frac{24 π - 7 π}{12}$

Step 12.4.3.2

Subtract $7 π$ from $24 π$ .

$x = \frac{17 π}{12}$

Step 12.5

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

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Step 12.5.1

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

$\frac{2 π}{| b |}$

Step 12.5.2

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

$\frac{2 π}{| 1 |}$

Step 12.5.3

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

$\frac{2 π}{1}$

Step 12.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 12.6

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

$x = \frac{7 π}{12} + 2 π n, \frac{17 π}{12} + 2 π n$ , for any integer $n$

Step 13

List all of the solutions.

$x = \frac{5 π}{12} + 2 π n, \frac{19 π}{12} + 2 π n, \frac{7 π}{12} + 2 π n, \frac{17 π}{12} + 2 π n$ , for any integer $n$

Step 14

Consolidate the solutions.

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Step 14.1

Consolidate $\frac{5 π}{12} + 2 π n$ and $\frac{17 π}{12} + 2 π n$ to $\frac{5 π}{12} + π n$ .

$x = \frac{5 π}{12} + π n, \frac{19 π}{12} + 2 π n, \frac{7 π}{12} + 2 π n$ , for any integer $n$

Step 14.2

Consolidate $\frac{19 π}{12} + 2 π n$ and $\frac{7 π}{12} + 2 π n$ to $\frac{7 π}{12} + π n$ .

$x = \frac{5 π}{12} + π n, \frac{7 π}{12} + π n$ , for any integer $n$