Trigonometry Examples

$\sin (x) \cos (2 x) + \cos (x) \sin (2 x) = 0$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Multiply $\sin (x)$ by $1$ .

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

Step 1.4

Rewrite using the commutative property of multiplication.

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

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

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

Step 1.5.3

Add $2$ and $1$ .

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

Step 1.6

Apply the sine double-angle identity.

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

Step 1.7

Rewrite using the commutative property of multiplication.

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

Step 1.8

Multiply $2 \cos (x) \sin (x) \cos (x)$ .

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Step 1.8.1

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

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.8.4

Add $1$ and $1$ .

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

Step 2

Factor $\sin (x)$ out of $\sin (x) - 2 \sin^{3} (x) + 2 \cos^{2} (x) \sin (x)$ .

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Step 2.1

Factor $\sin (x)$ out of $\sin^{1} (x)$ .

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

Step 2.2

Factor $\sin (x)$ out of $- 2 \sin^{3} (x)$ .

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

Step 2.3

Factor $\sin (x)$ out of $2 \cos^{2} (x) \sin (x)$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

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

$\sin (x) = 0$

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

Step 4

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

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Step 4.1

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

$\sin (x) = 0$

Step 4.2

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

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Step 4.2.1

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

$x = \arcsin (0)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

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

$x = 0$

Step 4.2.3

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 4.2.4

Subtract $0$ from $π$ .

$x = π$

Step 4.2.5

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

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Step 4.2.5.1

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

$\frac{2 π}{| b |}$

Step 4.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.2.5.3

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

$\frac{2 π}{1}$

Step 4.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.6

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 5

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

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Apply the distributive property.

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

Step 5.2.2.2

Multiply $- 2$ by $1$ .

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

Step 5.2.2.3

Multiply $- 1$ by $- 2$ .

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

Step 5.2.3

Simplify by adding terms.

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Step 5.2.3.1

Subtract $2$ from $1$ .

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

Step 5.2.3.2

Add $2 \cos^{2} (x)$ and $2 \cos^{2} (x)$ .

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

Step 5.2.4

Reorder the polynomial.

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

Step 5.2.5

Add $1$ to both sides of the equation.

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

Step 5.2.6

Divide each term in $4 \cos^{2} (x) = 1$ by $4$ and simplify.

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Step 5.2.6.1

Divide each term in $4 \cos^{2} (x) = 1$ by $4$ .

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

Step 5.2.6.2

Simplify the left side.

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Step 5.2.6.2.1

Cancel the common factor of $4$ .

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Step 5.2.6.2.1.1

Cancel the common factor.

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

Step 5.2.6.2.1.2

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

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

Step 5.2.7

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

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

Step 5.2.8

Simplify $\pm \sqrt{\frac{1}{4}}$ .

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Step 5.2.8.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

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

Step 5.2.8.2

Any root of $1$ is $1$ .

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

Step 5.2.8.3

Simplify the denominator.

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Step 5.2.8.3.1

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

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

Step 5.2.8.3.2

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

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

Step 5.2.9

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

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Step 5.2.9.1

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

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

Step 5.2.9.2

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

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

Step 5.2.9.3

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

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

Step 5.2.10

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

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

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

Step 5.2.11

Solve for $x$ in $\cos (x) = \frac{1}{2}$ .

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Step 5.2.11.1

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

$x = \arccos (\frac{1}{2})$

Step 5.2.11.2

Simplify the right side.

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Step 5.2.11.2.1

The exact value of $\arccos (\frac{1}{2})$ is $\frac{π}{3}$ .

$x = \frac{π}{3}$

Step 5.2.11.3

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.

$x = 2 π - \frac{π}{3}$

Step 5.2.11.4

Simplify $2 π - \frac{π}{3}$ .

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Step 5.2.11.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.2.11.4.2

Combine fractions.

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Step 5.2.11.4.2.1

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 5.2.11.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - π}{3}$

Step 5.2.11.4.3

Simplify the numerator.

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Step 5.2.11.4.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - π}{3}$

Step 5.2.11.4.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{3}$

Step 5.2.11.5

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

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Step 5.2.11.5.1

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

$\frac{2 π}{| b |}$

Step 5.2.11.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.11.5.3

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

$\frac{2 π}{1}$

Step 5.2.11.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.11.6

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

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

Step 5.2.12

Solve for $x$ in $\cos (x) = - \frac{1}{2}$ .

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Step 5.2.12.1

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

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

Step 5.2.12.2

Simplify the right side.

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Step 5.2.12.2.1

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

$x = \frac{2 π}{3}$

Step 5.2.12.3

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.

$x = 2 π - \frac{2 π}{3}$

Step 5.2.12.4

Simplify $2 π - \frac{2 π}{3}$ .

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Step 5.2.12.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.2.12.4.2

Combine fractions.

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Step 5.2.12.4.2.1

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 5.2.12.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - 2 π}{3}$

Step 5.2.12.4.3

Simplify the numerator.

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Step 5.2.12.4.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - 2 π}{3}$

Step 5.2.12.4.3.2

Subtract $2 π$ from $6 π$ .

$x = \frac{4 π}{3}$

Step 5.2.12.5

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

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Step 5.2.12.5.1

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

$\frac{2 π}{| b |}$

Step 5.2.12.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.12.5.3

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

$\frac{2 π}{1}$

Step 5.2.12.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.12.6

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

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

Step 5.2.13

List all of the solutions.

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

Step 5.2.14

Consolidate the solutions.

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Step 5.2.14.1

Consolidate $\frac{π}{3} + 2 π n$ and $\frac{4 π}{3} + 2 π n$ to $\frac{π}{3} + π n$ .

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

Step 5.2.14.2

Consolidate $\frac{5 π}{3} + 2 π n$ and $\frac{2 π}{3} + 2 π n$ to $\frac{2 π}{3} + π n$ .

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

Step 6

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

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

Step 7

Consolidate the answers.

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Step 7.1

Consolidate $2 π n$ and $π + 2 π n$ to $π n$ .

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

Step 7.2

Consolidate the answers.

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