Precalculus Examples

$\sqrt{3} \cos (2 x) = 2 \sin (x) \cos (2 x)$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify the left side.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

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

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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Step 4.1.3.2.1

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

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

Step 4.1.3.2.2

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

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

Step 4.1.3.3

Add $2$ and $1$ .

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

Step 4.1.4

Multiply $- 1$ by $- 2$ .

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

Step 5

Solve the equation for $x$ .

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Step 5.1

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

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Step 5.1.1

Factor out the greatest common factor from each group.

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Step 5.1.1.1

Group the first two terms and the last two terms.

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

Step 5.1.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 5.1.2

Factor the polynomial by factoring out the greatest common factor, $\cos^{2} (x) - \sin^{2} (x)$ .

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

Step 5.1.3

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

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

Step 5.2

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

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

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

$\sqrt{3} - 2 \sin (x) = 0$

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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Step 5.3.2.1

Divide each term in the equation by $\cos (x)$ .

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

Step 5.3.2.2

Cancel the common factor of $\cos (x)$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Rewrite the expression.

$1 + \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 5.3.2.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$1 + \tan (x) = \frac{0}{\cos (x)}$

Step 5.3.2.4

Separate fractions.

$1 + \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 5.3.2.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$1 + \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 5.3.2.6

Divide $0$ by $1$ .

$1 + \tan (x) = 0 \sec (x)$

Step 5.3.2.7

Multiply $0$ by $\sec (x)$ .

$1 + \tan (x) = 0$

Step 5.3.2.8

Subtract $1$ from both sides of the equation.

$\tan (x) = - 1$

Step 5.3.2.9

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

$x = \arctan (- 1)$

Step 5.3.2.10

Simplify the right side.

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Step 5.3.2.10.1

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$x = - \frac{π}{4}$

Step 5.3.2.11

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$x = - \frac{π}{4} - π$

Step 5.3.2.12

Simplify the expression to find the second solution.

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Step 5.3.2.12.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 5.3.2.12.2

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$x = \frac{3 π}{4}$

Step 5.3.2.13

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

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Step 5.3.2.13.1

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

$\frac{π}{| b |}$

Step 5.3.2.13.2

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

$\frac{π}{| 1 |}$

Step 5.3.2.13.3

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

$\frac{π}{1}$

Step 5.3.2.13.4

Divide $π$ by $1$ .

$π$

Step 5.3.2.14

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

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Step 5.3.2.14.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 5.3.2.14.2

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

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 5.3.2.14.3

Combine fractions.

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Step 5.3.2.14.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 5.3.2.14.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 5.3.2.14.4

Simplify the numerator.

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Step 5.3.2.14.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 5.3.2.14.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 5.3.2.14.5

List the new angles.

$x = \frac{3 π}{4}$

Step 5.3.2.15

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

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Step 5.4.2.1

Divide each term in the equation by $\cos (x)$ .

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

Step 5.4.2.2

Cancel the common factor of $\cos (x)$ .

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Step 5.4.2.2.1

Cancel the common factor.

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

Step 5.4.2.2.2

Rewrite the expression.

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

Step 5.4.2.3

Separate fractions.

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

Step 5.4.2.4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$1 + \frac{- 1}{1} \cdot \tan (x) = \frac{0}{\cos (x)}$

Step 5.4.2.5

Divide $- 1$ by $1$ .

$1 - \tan (x) = \frac{0}{\cos (x)}$

Step 5.4.2.6

Separate fractions.

$1 - \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 5.4.2.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$1 - \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 5.4.2.8

Divide $0$ by $1$ .

$1 - \tan (x) = 0 \sec (x)$

Step 5.4.2.9

Multiply $0$ by $\sec (x)$ .

$1 - \tan (x) = 0$

Step 5.4.2.10

Subtract $1$ from both sides of the equation.

$- \tan (x) = - 1$

Step 5.4.2.11

Divide each term in $- \tan (x) = - 1$ by $- 1$ and simplify.

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Step 5.4.2.11.1

Divide each term in $- \tan (x) = - 1$ by $- 1$ .

$\frac{- \tan (x)}{- 1} = \frac{- 1}{- 1}$

Step 5.4.2.11.2

Simplify the left side.

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Step 5.4.2.11.2.1

Dividing two negative values results in a positive value.

$\frac{\tan (x)}{1} = \frac{- 1}{- 1}$

Step 5.4.2.11.2.2

Divide $\tan (x)$ by $1$ .

$\tan (x) = \frac{- 1}{- 1}$

Step 5.4.2.11.3

Simplify the right side.

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Step 5.4.2.11.3.1

Divide $- 1$ by $- 1$ .

$\tan (x) = 1$

Step 5.4.2.12

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

$x = \arctan (1)$

Step 5.4.2.13

Simplify the right side.

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Step 5.4.2.13.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 5.4.2.14

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + \frac{π}{4}$

Step 5.4.2.15

Simplify $π + \frac{π}{4}$ .

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Step 5.4.2.15.1

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

$x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 5.4.2.15.2

Combine fractions.

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Step 5.4.2.15.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 5.4.2.15.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 5.4.2.15.3

Simplify the numerator.

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Step 5.4.2.15.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 5.4.2.15.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 5.4.2.16

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

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Step 5.4.2.16.1

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

$\frac{π}{| b |}$

Step 5.4.2.16.2

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

$\frac{π}{| 1 |}$

Step 5.4.2.16.3

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

$\frac{π}{1}$

Step 5.4.2.16.4

Divide $π$ by $1$ .

$π$

Step 5.4.2.17

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

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

Step 5.5

Set $\sqrt{3} - 2 \sin (x)$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $\sqrt{3} - 2 \sin (x)$ equal to $0$ .

$\sqrt{3} - 2 \sin (x) = 0$

Step 5.5.2

Solve $\sqrt{3} - 2 \sin (x) = 0$ for $x$ .

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Step 5.5.2.1

Subtract $\sqrt{3}$ from both sides of the equation.

$- 2 \sin (x) = - \sqrt{3}$

Step 5.5.2.2

Divide each term in $- 2 \sin (x) = - \sqrt{3}$ by $- 2$ and simplify.

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Step 5.5.2.2.1

Divide each term in $- 2 \sin (x) = - \sqrt{3}$ by $- 2$ .

$\frac{- 2 \sin (x)}{- 2} = \frac{- \sqrt{3}}{- 2}$

Step 5.5.2.2.2

Simplify the left side.

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Step 5.5.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 5.5.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 \sin (x)}{- 2} = \frac{- \sqrt{3}}{- 2}$

Step 5.5.2.2.2.1.2

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

$\sin (x) = \frac{- \sqrt{3}}{- 2}$

Step 5.5.2.2.3

Simplify the right side.

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Step 5.5.2.2.3.1

Dividing two negative values results in a positive value.

$\sin (x) = \frac{\sqrt{3}}{2}$

Step 5.5.2.3

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

$x = \arcsin (\frac{\sqrt{3}}{2})$

Step 5.5.2.4

Simplify the right side.

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Step 5.5.2.4.1

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

$x = \frac{π}{3}$

Step 5.5.2.5

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.

$x = π - \frac{π}{3}$

Step 5.5.2.6

Simplify $π - \frac{π}{3}$ .

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Step 5.5.2.6.1

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

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

Step 5.5.2.6.2

Combine fractions.

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Step 5.5.2.6.2.1

Combine $π$ and $\frac{3}{3}$ .

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

Step 5.5.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 - π}{3}$

Step 5.5.2.6.3

Simplify the numerator.

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Step 5.5.2.6.3.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π - π}{3}$

Step 5.5.2.6.3.2

Subtract $π$ from $3 π$ .

$x = \frac{2 π}{3}$

Step 5.5.2.7

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

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Step 5.5.2.7.1

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

$\frac{2 π}{| b |}$

Step 5.5.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 5.5.2.7.3

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

$\frac{2 π}{1}$

Step 5.5.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.5.2.8

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

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

Step 5.6

The final solution is all the values that make $(\cos (x) + \sin (x)) (\cos (x) - \sin (x)) (\sqrt{3} - 2 \sin (x)) = 0$ true.

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

Step 6

Consolidate the answers.

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Step 6.1

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

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

Step 6.2

Consolidate $\frac{π}{4} + \frac{π n}{2}$ and $\frac{5 π}{4} + π n$ to $\frac{π}{4} + \frac{π n}{2}$ .

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

Step 7