Trigonometry Examples

$\sin^{2} (x) \cos^{2} (x) = \frac{1}{4}$

Step 1

Subtract $\frac{1}{4}$ from both sides of the equation.

$\sin^{2} (x) \cos^{2} (x) - \frac{1}{4} = 0$

Step 2

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

$(1 - \cos^{2} (x)) \cos^{2} (x) - \frac{1}{4} = 0$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$1 \cos^{2} (x) - \cos^{2} (x) \cos^{2} (x) - \frac{1}{4} = 0$

Step 3.2

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

$\cos^{2} (x) - \cos^{2} (x) \cos^{2} (x) - \frac{1}{4} = 0$

Step 3.3

Multiply $\cos^{2} (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 3.3.1

Move $\cos^{2} (x)$ .

$\cos^{2} (x) - (\cos^{2} (x) \cos^{2} (x)) - \frac{1}{4} = 0$

Step 3.3.2

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

$\cos^{2} (x) - {\cos (x)}^{2 + 2} - \frac{1}{4} = 0$

Step 3.3.3

Add $2$ and $2$ .

$\cos^{2} (x) - \cos^{4} (x) - \frac{1}{4} = 0$

Step 4

Reorder the polynomial.

$- \cos^{4} (x) + \cos^{2} (x) - \frac{1}{4} = 0$

Step 5

Substitute $u = \cos^{2} (x)$ into the equation. This will make the quadratic formula easy to use.

$- u^{2} + u - \frac{1}{4} = 0$

$u = \cos^{2} (x)$

Step 6

Factor the left side of the equation.

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Step 6.1

Factor $- 1$ out of $- u^{2} + u - \frac{1}{4}$ .

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Step 6.1.1

Factor $- 1$ out of $- u^{2}$ .

$- (u^{2}) + u - \frac{1}{4} = 0$

Step 6.1.2

Factor $- 1$ out of $u$ .

$- (u^{2}) - 1 (- u) - \frac{1}{4} = 0$

Step 6.1.3

Factor $- 1$ out of $- \frac{1}{4}$ .

$- (u^{2}) - 1 (- u) - (\frac{1}{4}) = 0$

Step 6.1.4

Factor $- 1$ out of $- (u^{2}) - 1 (- u)$ .

$- (u^{2} - u) - (\frac{1}{4}) = 0$

Step 6.1.5

Factor $- 1$ out of $- (u^{2} - u) - (\frac{1}{4})$ .

$- (u^{2} - u + \frac{1}{4}) = 0$

Step 6.2

Factor using the perfect square rule.

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Step 6.2.1

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

$- (u^{2} - u + \frac{1^{2}}{4}) = 0$

Step 6.2.2

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

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

Step 6.2.3

Rewrite $\frac{1^{2}}{2^{2}}$ as ${(\frac{1}{2})}^{2}$ .

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

Step 6.2.4

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$u = 2 \cdot u \cdot \frac{1}{2}$

Step 6.2.5

Rewrite the polynomial.

$- (u^{2} - 2 \cdot u \cdot \frac{1}{2} + {(\frac{1}{2})}^{2}) = 0$

Step 6.2.6

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = \frac{1}{2}$ .

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

Step 7

Divide each term in $- {(u - \frac{1}{2})}^{2} = 0$ by $- 1$ and simplify.

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Step 7.1

Divide each term in $- {(u - \frac{1}{2})}^{2} = 0$ by $- 1$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Dividing two negative values results in a positive value.

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

Step 7.2.2

Divide ${(u - \frac{1}{2})}^{2}$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $0$ by $- 1$ .

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

Step 8

Set the $u - \frac{1}{2}$ equal to $0$ .

$u - \frac{1}{2} = 0$

Step 9

Add $\frac{1}{2}$ to both sides of the equation.

$u = \frac{1}{2}$

Step 10

Substitute the real value of $u = \cos^{2} (x)$ back into the solved equation.

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

Step 11

Solve the equation for $\cos (x)$ .

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Step 11.1

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

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

Step 11.2

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

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Step 11.2.1

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

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

Step 11.2.2

Any root of $1$ is $1$ .

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

Step 11.2.3

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

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

Step 11.2.4

Combine and simplify the denominator.

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Step 11.2.4.1

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

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

Step 11.2.4.2

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

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

Step 11.2.4.3

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

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

Step 11.2.4.4

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

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

Step 11.2.4.5

Add $1$ and $1$ .

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

Step 11.2.4.6

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

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Step 11.2.4.6.1

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

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

Step 11.2.4.6.2

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

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

Step 11.2.4.6.3

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

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

Step 11.2.4.6.4

Cancel the common factor of $2$ .

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Step 11.2.4.6.4.1

Cancel the common factor.

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

Step 11.2.4.6.4.2

Rewrite the expression.

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

Step 11.2.4.6.5

Evaluate the exponent.

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

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

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

Step 12

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

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

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

Step 13

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

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Step 13.1

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

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

Step 13.2

Simplify the right side.

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Step 13.2.1

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

$x = \frac{π}{4}$

Step 13.3

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.

$x = 2 π - \frac{π}{4}$

Step 13.4

Simplify $2 π - \frac{π}{4}$ .

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Step 13.4.1

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

$x = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 13.4.2

Combine fractions.

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Step 13.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 13.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 13.4.3

Simplify the numerator.

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Step 13.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 13.4.3.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 13.5

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

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Step 13.5.1

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

$\frac{2 π}{| b |}$

Step 13.5.2

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

$\frac{2 π}{| 1 |}$

Step 13.5.3

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

$\frac{2 π}{1}$

Step 13.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 13.6

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

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

Step 14

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

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Step 14.1

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

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

Step 14.2

Simplify the right side.

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Step 14.2.1

The exact value of $\arccos (- \frac{\sqrt{2}}{2})$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 14.3

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.

$x = 2 π - \frac{3 π}{4}$

Step 14.4

Simplify $2 π - \frac{3 π}{4}$ .

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Step 14.4.1

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

$x = 2 π \cdot \frac{4}{4} - \frac{3 π}{4}$

Step 14.4.2

Combine fractions.

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Step 14.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{3 π}{4}$

Step 14.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Step 14.4.3

Simplify the numerator.

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Step 14.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - 3 π}{4}$

Step 14.4.3.2

Subtract $3 π$ from $8 π$ .

$x = \frac{5 π}{4}$

Step 14.5

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

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Step 14.5.1

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

$\frac{2 π}{| b |}$

Step 14.5.2

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

$\frac{2 π}{| 1 |}$

Step 14.5.3

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

$\frac{2 π}{1}$

Step 14.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 14.6

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

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

Step 15

List all of the solutions.

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

Step 16

Consolidate the answers.

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