Trigonometry Examples

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

Step 1

Subtract $\sin (2 x)$ from both sides of the equation.

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

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 3

Simplify each side of the equation.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Multiply the exponents in ${({(2 \cos (x))}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Apply the product rule to $- \sin (2 x)$ .

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

Step 3.3.1.2

Raise $- 1$ to the power of $2$ .

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

Step 3.3.1.3

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

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

Step 4

Solve for $x$ .

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Step 4.1

Subtract $\sin^{2} (2 x)$ from both sides of the equation.

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

Step 4.2

Replace $\sin^{2} (2 x)$ with $1 - \cos^{2} (2 x)$ .

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

Step 4.3

Simplify the left side of the equation.

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Step 4.3.1

Apply pythagorean identity.

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

Step 4.3.2

Simplify each term.

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Step 4.3.2.1

Apply the sine double-angle identity.

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

Step 4.3.2.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.3.2.2.1

Apply the product rule to $2 \sin (x) \cos (x)$ .

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

Step 4.3.2.2.2

Apply the product rule to $2 \sin (x)$ .

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

Step 4.3.2.3

Raise $2$ to the power of $2$ .

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

Step 4.3.2.4

Multiply $4$ by $- 1$ .

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

Step 4.4

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

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Step 4.4.1

Factor $2 \cos (x)$ out of $2 \cos (x)$ .

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

Step 4.4.2

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

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

Step 4.4.3

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

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

Step 4.5

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

$\cos (x) = 0$

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

Step 4.6

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

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Step 4.6.1

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

$\cos (x) = 0$

Step 4.6.2

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

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Step 4.6.2.1

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

$x = \arccos (0)$

Step 4.6.2.2

Simplify the right side.

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Step 4.6.2.2.1

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

$x = \frac{π}{2}$

Step 4.6.2.3

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 4.6.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 4.6.2.4.1

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 4.6.2.4.2

Combine fractions.

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Step 4.6.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 4.6.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 4.6.2.4.3

Simplify the numerator.

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Step 4.6.2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 4.6.2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 4.6.2.5

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

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Step 4.6.2.5.1

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

$\frac{2 π}{| b |}$

Step 4.6.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.6.2.5.3

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

$\frac{2 π}{1}$

Step 4.6.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.6.2.6

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 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

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

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Step 4.7.2.1

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

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

Step 4.7.2.2

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

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

Step 4.7.2.3

Apply the distributive property.

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

Step 4.7.2.4

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

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

Step 4.7.2.5

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

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Step 4.7.2.5.1

Move $\cos (x)$ .

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

Step 4.7.2.5.2

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

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Step 4.7.2.5.2.1

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

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

Step 4.7.2.5.2.2

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

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

Step 4.7.2.5.3

Add $1$ and $2$ .

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

Step 4.7.2.6

Reorder the polynomial.

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

Step 4.7.2.7

Substitute $u$ for $\cos (x)$ .

$- {(u)}^{3} + u = 0$

Step 4.7.2.8

Factor the left side of the equation.

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Step 4.7.2.8.1

Factor $- u$ out of $- u^{3} + u$ .

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Step 4.7.2.8.1.1

Factor $- u$ out of $- u^{3}$ .

$- u \cdot u^{2} + u = 0$

Step 4.7.2.8.1.2

Rewrite $u$ as $1 u$ .

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

Step 4.7.2.8.1.3

Factor $- u$ out of $1 u$ .

$- u \cdot u^{2} - u \cdot - 1 = 0$

Step 4.7.2.8.1.4

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

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

Step 4.7.2.8.2

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

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

Step 4.7.2.8.3

Factor.

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Step 4.7.2.8.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = u$ and $b = 1$ .

$- u ((u + 1) (u - 1)) = 0$

Step 4.7.2.8.3.2

Remove unnecessary parentheses.

$- u (u + 1) (u - 1) = 0$

Step 4.7.2.9

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

$u = 0$

$u + 1 = 0$

$u - 1 = 0$

Step 4.7.2.10

Set $u$ equal to $0$ .

$u = 0$

Step 4.7.2.11

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 4.7.2.11.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 4.7.2.11.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 4.7.2.12

Set $u - 1$ equal to $0$ and solve for $u$ .

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Step 4.7.2.12.1

Set $u - 1$ equal to $0$ .

$u - 1 = 0$

Step 4.7.2.12.2

Add $1$ to both sides of the equation.

$u = 1$

Step 4.7.2.13

The final solution is all the values that make $- u (u + 1) (u - 1) = 0$ true.

$u = 0, - 1, 1$

Step 4.7.2.14

Substitute $\cos (x)$ for $u$ .

$\cos (x) = 0, - 1, 1$

Step 4.7.2.15

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

$\cos (x) = 0$

$\cos (x) = - 1$

$\cos (x) = 1$

Step 4.7.2.16

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

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Step 4.7.2.16.1

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

$x = \arccos (0)$

Step 4.7.2.16.2

Simplify the right side.

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Step 4.7.2.16.2.1

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

$x = \frac{π}{2}$

Step 4.7.2.16.3

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 4.7.2.16.4

Simplify $2 π - \frac{π}{2}$ .

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Step 4.7.2.16.4.1

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 4.7.2.16.4.2

Combine fractions.

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Step 4.7.2.16.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 4.7.2.16.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 4.7.2.16.4.3

Simplify the numerator.

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Step 4.7.2.16.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 4.7.2.16.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 4.7.2.16.5

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

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Step 4.7.2.16.5.1

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

$\frac{2 π}{| b |}$

Step 4.7.2.16.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.7.2.16.5.3

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

$\frac{2 π}{1}$

Step 4.7.2.16.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.7.2.16.6

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 4.7.2.17

Solve for $x$ in $\cos (x) = - 1$ .

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Step 4.7.2.17.1

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

$x = \arccos (- 1)$

Step 4.7.2.17.2

Simplify the right side.

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Step 4.7.2.17.2.1

The exact value of $\arccos (- 1)$ is $π$ .

$x = π$

Step 4.7.2.17.3

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.

$x = 2 π - π$

Step 4.7.2.17.4

Subtract $π$ from $2 π$ .

$x = π$

Step 4.7.2.17.5

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

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Step 4.7.2.17.5.1

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

$\frac{2 π}{| b |}$

Step 4.7.2.17.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.7.2.17.5.3

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

$\frac{2 π}{1}$

Step 4.7.2.17.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.7.2.17.6

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

$x = π + 2 π n$ , for any integer $n$

Step 4.7.2.18

Solve for $x$ in $\cos (x) = 1$ .

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Step 4.7.2.18.1

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

$x = \arccos (1)$

Step 4.7.2.18.2

Simplify the right side.

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Step 4.7.2.18.2.1

The exact value of $\arccos (1)$ is $0$ .

$x = 0$

Step 4.7.2.18.3

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.

$x = 2 π - 0$

Step 4.7.2.18.4

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 4.7.2.18.5

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

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Step 4.7.2.18.5.1

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

$\frac{2 π}{| b |}$

Step 4.7.2.18.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.7.2.18.5.3

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

$\frac{2 π}{1}$

Step 4.7.2.18.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.7.2.18.6

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

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

Step 4.7.2.19

List all of the solutions.

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

Step 4.7.2.20

Consolidate the answers.

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

Step 4.8

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

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

Step 5

Consolidate the answers.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Verify each of the solutions by substituting them into $\sin (2 x) + \sqrt{2 \cos (x)} = 0$ and solving.

$x = 4.71238898, 10.99557428$ , for any integer $n$