Precalculus Examples

$\sin (x) \cos (3 x) + \cos (x) \sin (3 x) = 0$

Step 1

Simplify the left side of the equation.

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Step 1.1

Simplify each term.

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Step 1.1.1

Use the triple-angle identity to transform $\cos (3 x)$ to $4 \cos^{3} (x) - 3 \cos (x)$ .

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

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

Rewrite using the commutative property of multiplication.

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

Step 1.1.4

Rewrite using the commutative property of multiplication.

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

Step 1.1.5

Apply the sine triple-angle identity.

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

Step 1.1.6

Apply the distributive property.

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

Step 1.1.7

Rewrite using the commutative property of multiplication.

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

Step 1.1.8

Rewrite using the commutative property of multiplication.

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

Step 1.2

Combine the opposite terms in $4 \sin (x) \cos^{3} (x) - 3 \sin (x) \cos (x) - 4 \cos (x) \sin^{3} (x) + 3 \cos (x) \sin (x)$ .

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Step 1.2.1

Reorder the factors in the terms $- 3 \sin (x) \cos (x)$ and $3 \cos (x) \sin (x)$ .

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

Step 1.2.2

Add $- 3 \cos (x) \sin (x)$ and $3 \cos (x) \sin (x)$ .

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

Step 1.2.3

Add $4 \sin (x) \cos^{3} (x) - 4 \cos (x) \sin^{3} (x)$ and $0$ .

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

Step 2

Factor $4 \sin (x) \cos^{3} (x) - 4 \cos (x) \sin^{3} (x)$ .

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

Factor.

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Step 2.2.1

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 $b = \sin (x)$ .

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

Step 2.2.2

Remove unnecessary parentheses.

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

Step 3

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

$\sin (x) = 0$

$\cos (x) = 0$

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

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

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

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Step 5.1

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

$\cos (x) = 0$

Step 5.2

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

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Step 5.2.1

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

$x = \arccos (0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

$x = \frac{π}{2}$

Step 5.2.3

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 5.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 5.2.4.1

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 5.2.4.2

Combine fractions.

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Step 5.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 5.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 5.2.4.3

Simplify the numerator.

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Step 5.2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 5.2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 5.2.5

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

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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 $\cos (x)$ 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 6

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

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

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

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Step 6.2.2.1

Cancel the common factor.

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

Step 6.2.2.2

Rewrite the expression.

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

Step 6.2.3

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

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

Step 6.2.4

Separate fractions.

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

Step 6.2.5

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

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

Step 6.2.6

Divide $0$ by $1$ .

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

Step 6.2.7

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

$1 + \tan (x) = 0$

Step 6.2.8

Subtract $1$ from both sides of the equation.

$\tan (x) = - 1$

Step 6.2.9

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

$x = \arctan (- 1)$

Step 6.2.10

Simplify the right side.

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Step 6.2.10.1

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

$x = - \frac{π}{4}$

Step 6.2.11

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 6.2.12

Simplify the expression to find the second solution.

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Step 6.2.12.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 6.2.12.2

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

$x = \frac{3 π}{4}$

Step 6.2.13

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

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Step 6.2.13.1

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

$\frac{π}{| b |}$

Step 6.2.13.2

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

$\frac{π}{| 1 |}$

Step 6.2.13.3

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

$\frac{π}{1}$

Step 6.2.13.4

Divide $π$ by $1$ .

$π$

Step 6.2.14

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

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Step 6.2.14.1

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

$- \frac{π}{4} + π$

Step 6.2.14.2

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

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 6.2.14.3

Combine fractions.

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Step 6.2.14.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 6.2.14.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 6.2.14.4

Simplify the numerator.

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Step 6.2.14.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 6.2.14.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 6.2.14.5

List the new angles.

$x = \frac{3 π}{4}$

Step 6.2.15

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 7

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

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Step 7.1

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

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

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

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

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Step 7.2.2.1

Cancel the common factor.

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

Step 7.2.2.2

Rewrite the expression.

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

Step 7.2.3

Separate fractions.

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

Step 7.2.4

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

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

Step 7.2.5

Divide $- 1$ by $1$ .

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

Step 7.2.6

Separate fractions.

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

Step 7.2.7

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

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

Step 7.2.8

Divide $0$ by $1$ .

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

Step 7.2.9

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

$1 - \tan (x) = 0$

Step 7.2.10

Subtract $1$ from both sides of the equation.

$- \tan (x) = - 1$

Step 7.2.11

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

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Step 7.2.11.1

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

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

Step 7.2.11.2

Simplify the left side.

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Step 7.2.11.2.1

Dividing two negative values results in a positive value.

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

Step 7.2.11.2.2

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

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

Step 7.2.11.3

Simplify the right side.

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Step 7.2.11.3.1

Divide $- 1$ by $- 1$ .

$\tan (x) = 1$

Step 7.2.12

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

$x = \arctan (1)$

Step 7.2.13

Simplify the right side.

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Step 7.2.13.1

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

$x = \frac{π}{4}$

Step 7.2.14

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 7.2.15

Simplify $π + \frac{π}{4}$ .

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Step 7.2.15.1

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 7.2.15.2

Combine fractions.

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Step 7.2.15.2.1

Combine $π$ and $\frac{4}{4}$ .

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

Step 7.2.15.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 7.2.15.3

Simplify the numerator.

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Step 7.2.15.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 7.2.15.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 7.2.16

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

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Step 7.2.16.1

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

$\frac{π}{| b |}$

Step 7.2.16.2

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

$\frac{π}{| 1 |}$

Step 7.2.16.3

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

$\frac{π}{1}$

Step 7.2.16.4

Divide $π$ by $1$ .

$π$

Step 7.2.17

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 8

The final solution is all the values that make $4 \sin (x) \cos (x) (\cos (x) + \sin (x)) (\cos (x) - \sin (x)) = 0$ true.

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

Step 9

Consolidate the answers.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

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

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

Step 9.6

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

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