Precalculus Examples

$\cos (2 x) - \sqrt{3} \sin (x) = 1$

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

Simplify the left side of the equation.

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Step 2.1

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

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

Step 2.2

Combine the opposite terms in $1 - 2 \sin^{2} (x) - \sqrt{3} \sin (x) - 1$ .

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Step 2.2.1

Subtract $1$ from $1$ .

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

Step 2.2.2

Subtract $2 \sin^{2} (x)$ from $0$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

$\sin (x) = 0$

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

Step 5

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

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Step 5.1

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

$\sin (x) = 0$

Step 5.2

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

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Step 5.2.1

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

$x = \arcsin (0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

$x = 0$

Step 5.2.3

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 5.2.4

Subtract $0$ from $π$ .

$x = π$

Step 5.2.5

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

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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 $\sin (x)$ 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 6

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

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

Add $\sqrt{3}$ to both sides of the equation.

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

Step 6.2.2

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

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Step 6.2.2.1

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

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

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.2.1.2

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

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

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

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

Step 6.2.3

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

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

Step 6.2.4

Simplify the right side.

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Step 6.2.4.1

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

$x = - \frac{π}{3}$

Step 6.2.5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{3} + π$

Step 6.2.6

Simplify the expression to find the second solution.

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Step 6.2.6.1

Subtract $2 π$ from $2 π + \frac{π}{3} + π$ .

$x = 2 π + \frac{π}{3} + π - 2 π$

Step 6.2.6.2

The resulting angle of $\frac{4 π}{3}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{3} + π$ .

$x = \frac{4 π}{3}$

Step 6.2.7

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

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Step 6.2.7.1

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

$\frac{2 π}{| b |}$

Step 6.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 6.2.7.3

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

$\frac{2 π}{1}$

Step 6.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.2.8

Add $2 π$ to every negative angle to get positive angles.

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Step 6.2.8.1

Add $2 π$ to $- \frac{π}{3}$ to find the positive angle.

$- \frac{π}{3} + 2 π$

Step 6.2.8.2

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

$2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 6.2.8.3

Combine fractions.

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Step 6.2.8.3.1

Combine $2 π$ and $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 6.2.8.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - π}{3}$

Step 6.2.8.4

Simplify the numerator.

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Step 6.2.8.4.1

Multiply $3$ by $2$ .

$\frac{6 π - π}{3}$

Step 6.2.8.4.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{3}$

Step 6.2.8.5

List the new angles.

$x = \frac{5 π}{3}$

Step 6.2.9

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

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

Step 7

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

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

Step 8

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

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