Trigonometry Examples

$8 \sin (x) \cos (x) = \sin (4 x)$

Step 1

Subtract $\sin (4 x)$ from both sides of the equation.

$8 \sin (x) \cos (x) - \sin (4 x) = 0$

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

Factor $2$ out of $4 x$ .

$8 \sin (x) \cos (x) - \sin (2 (2 x)) = 0$

Step 2.1.2

Apply the sine double-angle identity.

$8 \sin (x) \cos (x) - (2 (2 \sin (x) \cos (x)) \cos (2 x)) = 0$

Step 2.1.3

Multiply $2$ by $2$ .

$8 \sin (x) \cos (x) - (4 (\sin (x) \cos (x)) \cos (2 x)) = 0$

Step 2.1.4

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

$8 \sin (x) \cos (x) - (4 \sin (x) \cos (x) (1 - 2 \sin^{2} (x))) = 0$

Step 2.1.5

Apply the distributive property.

$8 \sin (x) \cos (x) - (4 \sin (x) \cos (x) \cdot 1 + 4 \sin (x) \cos (x) (- 2 \sin^{2} (x))) = 0$

Step 2.1.6

Multiply $4$ by $1$ .

$8 \sin (x) \cos (x) - (4 \sin (x) \cos (x) + 4 \sin (x) \cos (x) (- 2 \sin^{2} (x))) = 0$

Step 2.1.7

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

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Step 2.1.7.1

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

$8 \sin (x) \cos (x) - (4 \sin (x) \cos (x) + 4 (\sin^{2} (x) \sin (x)) \cos (x) \cdot - 2) = 0$

Step 2.1.7.2

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

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Step 2.1.7.2.1

Raise $\sin (x)$ to the power of $1$ .

$8 \sin (x) \cos (x) - (4 \sin (x) \cos (x) + 4 (\sin^{2} (x) \sin (x)) \cos (x) \cdot - 2) = 0$

Step 2.1.7.2.2

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

$8 \sin (x) \cos (x) - (4 \sin (x) \cos (x) + 4 {\sin (x)}^{2 + 1} \cos (x) \cdot - 2) = 0$

Step 2.1.7.3

Add $2$ and $1$ .

$8 \sin (x) \cos (x) - (4 \sin (x) \cos (x) + 4 \sin^{3} (x) \cos (x) \cdot - 2) = 0$

Step 2.1.8

Multiply $- 2$ by $4$ .

$8 \sin (x) \cos (x) - (4 \sin (x) \cos (x) - 8 \sin^{3} (x) \cos (x)) = 0$

Step 2.1.9

Apply the distributive property.

$8 \sin (x) \cos (x) - (4 \sin (x) \cos (x)) - (- 8 \sin^{3} (x) \cos (x)) = 0$

Step 2.1.10

Multiply $4$ by $- 1$ .

$8 \sin (x) \cos (x) - 4 (\sin (x) \cos (x)) - (- 8 \sin^{3} (x) \cos (x)) = 0$

Step 2.1.11

Multiply $- 8$ by $- 1$ .

$8 \sin (x) \cos (x) - 4 (\sin (x) \cos (x)) + 8 (\sin^{3} (x) \cos (x)) = 0$

Step 2.1.12

Remove parentheses.

$8 \sin (x) \cos (x) - 4 \sin (x) \cos (x) + 8 \sin^{3} (x) \cos (x) = 0$

Step 2.2

Subtract $4 \sin (x) \cos (x)$ from $8 \sin (x) \cos (x)$ .

$4 \sin (x) \cos (x) + 8 \sin^{3} (x) \cos (x) = 0$

Step 3

Factor $4 \sin (x) \cos (x)$ out of $4 \sin (x) \cos (x) + 8 \sin^{3} (x) \cos (x)$ .

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Step 3.1

Factor $4 \sin (x) \cos (x)$ out of $4 \sin (x) \cos (x)$ .

$4 \sin (x) \cos (x) \cdot 1 + 8 \sin^{3} (x) \cos (x) = 0$

Step 3.2

Factor $4 \sin (x) \cos (x)$ out of $8 \sin^{3} (x) \cos (x)$ .

$4 \sin (x) \cos (x) \cdot 1 + 4 \sin (x) \cos (x) (2 \sin^{2} (x)) = 0$

Step 3.3

Factor $4 \sin (x) \cos (x)$ out of $4 \sin (x) \cos (x) \cdot 1 + 4 \sin (x) \cos (x) (2 \sin^{2} (x))$ .

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

Step 4

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

$\sin (x) = 0$

$\cos (x) = 0$

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

Step 5

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

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Step 5.1

Set $\sin (x)$ equal to $0$ .

$\sin (x) = 0$

Step 5.2

Solve $\sin (x) = 0$ for $x$ .

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Step 5.2.1

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

$x = \arcsin (0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

The exact value of $\arcsin (0)$ is $0$ .

$x = 0$

Step 5.2.3

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.

$x = π - 0$

Step 5.2.4

Subtract $0$ from $π$ .

$x = π$

Step 5.2.5

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

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Step 5.2.5.1

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

$\frac{2 π}{| b |}$

Step 5.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.5.3

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

$\frac{2 π}{1}$

Step 5.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.6

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

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

Step 6

Set $\cos (x)$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $\cos (x)$ equal to $0$ .

$\cos (x) = 0$

Step 6.2

Solve $\cos (x) = 0$ for $x$ .

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Step 6.2.1

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

$x = \arccos (0)$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 6.2.3

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.

$x = 2 π - \frac{π}{2}$

Step 6.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 6.2.4.1

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

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

Step 6.2.4.2

Combine fractions.

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Step 6.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 6.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 6.2.4.3

Simplify the numerator.

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Step 6.2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 6.2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 6.2.5

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

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Step 6.2.5.1

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

$\frac{2 π}{| b |}$

Step 6.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 6.2.5.3

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

$\frac{2 π}{1}$

Step 6.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.2.6

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

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

Step 7

Set $1 + 2 \sin^{2} (x)$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $1 + 2 \sin^{2} (x)$ equal to $0$ .

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

Step 7.2

Solve $1 + 2 \sin^{2} (x) = 0$ for $x$ .

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Step 7.2.1

Subtract $1$ from both sides of the equation.

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

Step 7.2.2

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

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Step 7.2.2.1

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

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

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.2.1.2

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

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

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative in front of the fraction.

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

Step 7.2.3

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

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

Step 7.2.4

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

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Step 7.2.4.1

Rewrite $- \frac{1}{2}$ as $i^{2} \frac{1^{2}}{2}$ .

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Step 7.2.4.1.1

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

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

Step 7.2.4.1.2

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

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

Step 7.2.4.2

Pull terms out from under the radical.

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

Step 7.2.4.3

One to any power is one.

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

Step 7.2.4.4

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

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

Step 7.2.4.5

Any root of $1$ is $1$ .

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

Step 7.2.4.6

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

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

Step 7.2.4.7

Combine and simplify the denominator.

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Step 7.2.4.7.1

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

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

Step 7.2.4.7.2

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

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

Step 7.2.4.7.3

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

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

Step 7.2.4.7.4

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

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

Step 7.2.4.7.5

Add $1$ and $1$ .

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

Step 7.2.4.7.6

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

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Step 7.2.4.7.6.1

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

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

Step 7.2.4.7.6.2

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

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

Step 7.2.4.7.6.3

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

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

Step 7.2.4.7.6.4

Cancel the common factor of $2$ .

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Step 7.2.4.7.6.4.1

Cancel the common factor.

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

Step 7.2.4.7.6.4.2

Rewrite the expression.

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

Step 7.2.4.7.6.5

Evaluate the exponent.

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

Step 7.2.4.8

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

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

Step 7.2.5

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

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Step 7.2.5.1

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

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

Step 7.2.5.2

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

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

Step 7.2.5.3

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

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

Step 7.2.6

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

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

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

Step 7.2.7

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

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Step 7.2.7.1

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

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

Step 7.2.7.2

The inverse sine of $\arcsin (\frac{i \sqrt{2}}{2})$ is undefined.

Undefined

Step 7.2.8

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

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Step 7.2.8.1

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

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

Step 7.2.8.2

The inverse sine of $\arcsin (- \frac{i \sqrt{2}}{2})$ is undefined.

Undefined

Step 7.2.9

List all of the solutions.

No solution

Step 8

The final solution is all the values that make $4 \sin (x) \cos (x) (1 + 2 \sin^{2} (x)) = 0$ true.

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

Step 9

Consolidate the answers.

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