Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Reorder the polynomial.

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

Step 4

Subtract $1$ from both sides of the equation.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

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

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Dividing two negative values results in a positive value.

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Any root of $1$ is $1$ .

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

Step 7.3

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

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

Step 7.4

Combine and simplify the denominator.

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Step 7.4.1

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

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

Step 7.4.2

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

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

Step 7.4.3

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

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

Step 7.4.4

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

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

Step 7.4.5

Add $1$ and $1$ .

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

Step 7.4.6

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

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Step 7.4.6.1

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

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

Step 7.4.6.2

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

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

Step 7.4.6.3

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

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

Step 7.4.6.4

Cancel the common factor of $2$ .

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Step 7.4.6.4.1

Cancel the common factor.

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

Step 7.4.6.4.2

Rewrite the expression.

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

Step 7.4.6.5

Evaluate the exponent.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

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

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

Simplify the right side.

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Step 10.2.1

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

$2 x = \frac{π}{4}$

Step 10.3

Divide each term in $2 x = \frac{π}{4}$ by $2$ and simplify.

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Step 10.3.1

Divide each term in $2 x = \frac{π}{4}$ by $2$ .

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

Step 10.3.2

Simplify the left side.

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Step 10.3.2.1

Cancel the common factor of $2$ .

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Step 10.3.2.1.1

Cancel the common factor.

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

Step 10.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{4}}{2}$

Step 10.3.3

Simplify the right side.

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Step 10.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{4} \cdot \frac{1}{2}$

Step 10.3.3.2

Multiply $\frac{π}{4} \cdot \frac{1}{2}$ .

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Step 10.3.3.2.1

Multiply $\frac{π}{4}$ by $\frac{1}{2}$ .

$x = \frac{π}{4 \cdot 2}$

Step 10.3.3.2.2

Multiply $4$ by $2$ .

$x = \frac{π}{8}$

Step 10.4

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.

$2 x = π - \frac{π}{4}$

Step 10.5

Solve for $x$ .

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Step 10.5.1

Simplify.

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Step 10.5.1.1

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

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

Step 10.5.1.2

Combine $π$ and $\frac{4}{4}$ .

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

Step 10.5.1.3

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 4 - π}{4}$

Step 10.5.1.4

Subtract $π$ from $π \cdot 4$ .

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Step 10.5.1.4.1

Reorder $π$ and $4$ .

$2 x = \frac{4 \cdot π - π}{4}$

Step 10.5.1.4.2

Subtract $π$ from $4 \cdot π$ .

$2 x = \frac{3 \cdot π}{4}$

Step 10.5.2

Divide each term in $2 x = \frac{3 \cdot π}{4}$ by $2$ and simplify.

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Step 10.5.2.1

Divide each term in $2 x = \frac{3 \cdot π}{4}$ by $2$ .

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

Step 10.5.2.2

Simplify the left side.

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Step 10.5.2.2.1

Cancel the common factor of $2$ .

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Step 10.5.2.2.1.1

Cancel the common factor.

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

Step 10.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 \cdot π}{4}}{2}$

Step 10.5.2.3

Simplify the right side.

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Step 10.5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 \cdot π}{4} \cdot \frac{1}{2}$

Step 10.5.2.3.2

Multiply $\frac{3 π}{4} \cdot \frac{1}{2}$ .

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Step 10.5.2.3.2.1

Multiply $\frac{3 π}{4}$ by $\frac{1}{2}$ .

$x = \frac{3 π}{4 \cdot 2}$

Step 10.5.2.3.2.2

Multiply $4$ by $2$ .

$x = \frac{3 π}{8}$

Step 10.6

Find the period of $\sin (2 x)$ .

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Step 10.6.1

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

$\frac{2 π}{| b |}$

Step 10.6.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 10.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 10.6.4

Cancel the common factor of $2$ .

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Step 10.6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 10.6.4.2

Divide $π$ by $1$ .

$π$

Step 10.7

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

$x = \frac{π}{8} + π n, \frac{3 π}{8} + π n$ , for any integer $n$

Step 11

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

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Step 11.1

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

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

Step 11.2

Simplify the right side.

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Step 11.2.1

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

$2 x = - \frac{π}{4}$

Step 11.3

Divide each term in $2 x = - \frac{π}{4}$ by $2$ and simplify.

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Step 11.3.1

Divide each term in $2 x = - \frac{π}{4}$ by $2$ .

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

Step 11.3.2

Simplify the left side.

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Step 11.3.2.1

Cancel the common factor of $2$ .

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Step 11.3.2.1.1

Cancel the common factor.

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

Step 11.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{4}}{2}$

Step 11.3.3

Simplify the right side.

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Step 11.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{4} \cdot \frac{1}{2}$

Step 11.3.3.2

Multiply $- \frac{π}{4} \cdot \frac{1}{2}$ .

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Step 11.3.3.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{4}$ .

$x = - \frac{π}{2 \cdot 4}$

Step 11.3.3.2.2

Multiply $2$ by $4$ .

$x = - \frac{π}{8}$

Step 11.4

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.

$2 x = 2 π + \frac{π}{4} + π$

Step 11.5

Simplify the expression to find the second solution.

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Step 11.5.1

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

$2 x = 2 π + \frac{π}{4} + π - 2 π$

Step 11.5.2

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

$2 x = \frac{5 π}{4}$

Step 11.5.3

Divide each term in $2 x = \frac{5 π}{4}$ by $2$ and simplify.

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Step 11.5.3.1

Divide each term in $2 x = \frac{5 π}{4}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{5 π}{4}}{2}$

Step 11.5.3.2

Simplify the left side.

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Step 11.5.3.2.1

Cancel the common factor of $2$ .

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Step 11.5.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{5 π}{4}}{2}$

Step 11.5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 π}{4}}{2}$

Step 11.5.3.3

Simplify the right side.

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Step 11.5.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 π}{4} \cdot \frac{1}{2}$

Step 11.5.3.3.2

Multiply $\frac{5 π}{4} \cdot \frac{1}{2}$ .

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Step 11.5.3.3.2.1

Multiply $\frac{5 π}{4}$ by $\frac{1}{2}$ .

$x = \frac{5 π}{4 \cdot 2}$

Step 11.5.3.3.2.2

Multiply $4$ by $2$ .

$x = \frac{5 π}{8}$

Step 11.6

Find the period of $\sin (2 x)$ .

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Step 11.6.1

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

$\frac{2 π}{| b |}$

Step 11.6.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 11.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 11.6.4

Cancel the common factor of $2$ .

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Step 11.6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 11.6.4.2

Divide $π$ by $1$ .

$π$

Step 11.7

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

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Step 11.7.1

Add $π$ to $- \frac{π}{8}$ to find the positive angle.

$- \frac{π}{8} + π$

Step 11.7.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$π \cdot \frac{8}{8} - \frac{π}{8}$

Step 11.7.3

Combine fractions.

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Step 11.7.3.1

Combine $π$ and $\frac{8}{8}$ .

$\frac{π \cdot 8}{8} - \frac{π}{8}$

Step 11.7.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 8 - π}{8}$

Step 11.7.4

Simplify the numerator.

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Step 11.7.4.1

Move $8$ to the left of $π$ .

$\frac{8 \cdot π - π}{8}$

Step 11.7.4.2

Subtract $π$ from $8 π$ .

$\frac{7 π}{8}$

Step 11.7.5

List the new angles.

$x = \frac{7 π}{8}$

Step 11.8

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

$x = \frac{5 π}{8} + π n, \frac{7 π}{8} + π n$ , for any integer $n$

Step 12

List all of the solutions.

$x = \frac{π}{8} + π n, \frac{3 π}{8} + π n, \frac{5 π}{8} + π n, \frac{7 π}{8} + π n$ , for any integer $n$

Step 13

Consolidate the answers.

$x = \frac{π}{8} + \frac{π n}{4}$ , for any integer $n$