Precalculus Examples

$2 \tan (x) \cdot \cos (x) - \tan (x) = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.1.1.1

Add parentheses.

$2 (\tan (x) \cdot \cos (x)) - \tan (x) = 0$

Step 1.1.1.2

Rewrite $2 \tan (x) \cdot \cos (x)$ in terms of sines and cosines.

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

Step 1.1.1.3

Cancel the common factors.

$2 \sin (x) - \tan (x) = 0$

Step 1.1.2

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 2

Multiply both sides of the equation by $\cos (x)$ .

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

Step 3

Apply the distributive property.

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

Step 4

Rewrite using the commutative property of multiplication.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

Simplify each term.

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Step 6.1

Reorder $2 \cos (x)$ and $\sin (x)$ .

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

Step 6.2

Reorder $\sin (x)$ and $2$ .

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

Step 6.3

Apply the sine double-angle identity.

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

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

Cancel the common factor.

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

Step 6.4.3

Rewrite the expression.

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

Step 7

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

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

Step 8

Apply the sine double-angle identity.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Factor $\sin (x)$ out of $- \sin (x)$ .

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

Step 9.3

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

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

Step 10

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

$\sin (x) = 0$

$2 \cos (x) - 1 = 0$

Step 11

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

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Step 11.1

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

$\sin (x) = 0$

Step 11.2

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

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Step 11.2.1

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

$x = \arcsin (0)$

Step 11.2.2

Simplify the right side.

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Step 11.2.2.1

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

$x = 0$

Step 11.2.3

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 11.2.4

Subtract $0$ from $π$ .

$x = π$

Step 11.2.5

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

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Step 11.2.5.1

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

$\frac{2 π}{| b |}$

Step 11.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 11.2.5.3

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

$\frac{2 π}{1}$

Step 11.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 11.2.6

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 12

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

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Step 12.1

Set $2 \cos (x) - 1$ equal to $0$ .

$2 \cos (x) - 1 = 0$

Step 12.2

Solve $2 \cos (x) - 1 = 0$ for $x$ .

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Step 12.2.1

Add $1$ to both sides of the equation.

$2 \cos (x) = 1$

Step 12.2.2

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

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Step 12.2.2.1

Divide each term in $2 \cos (x) = 1$ by $2$ .

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

Step 12.2.2.2

Simplify the left side.

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Step 12.2.2.2.1

Cancel the common factor of $2$ .

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Step 12.2.2.2.1.1

Cancel the common factor.

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

Step 12.2.2.2.1.2

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

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

Step 12.2.3

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

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

Step 12.2.4

Simplify the right side.

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Step 12.2.4.1

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

$x = \frac{π}{3}$

Step 12.2.5

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 12.2.6

Simplify $2 π - \frac{π}{3}$ .

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Step 12.2.6.1

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 12.2.6.2

Combine fractions.

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Step 12.2.6.2.1

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 12.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - π}{3}$

Step 12.2.6.3

Simplify the numerator.

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Step 12.2.6.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - π}{3}$

Step 12.2.6.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{3}$

Step 12.2.7

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

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Step 12.2.7.1

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

$\frac{2 π}{| b |}$

Step 12.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 12.2.7.3

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

$\frac{2 π}{1}$

Step 12.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 12.2.8

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 13

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

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

Step 14

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

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