Calculus Examples

$3 \sin^{2} (x) = \cos^{2} (x)$ , $0 < x < 2 π$

Step 1

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

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

Step 2

Replace the $3 \sin^{2} (x)$ with $3 (1 - \cos^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $3$ by $1$ .

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

Step 3.3

Multiply $- 1$ by $3$ .

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

Step 4

Subtract $\cos^{2} (x)$ from $- 3 \cos^{2} (x)$ .

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

Step 5

Reorder the polynomial.

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

Step 6

Subtract $3$ from both sides of the equation.

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

Step 7

Divide each term in $- 4 \cos^{2} (x) = - 3$ by $- 4$ and simplify.

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Step 7.1

Divide each term in $- 4 \cos^{2} (x) = - 3$ by $- 4$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 4$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

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

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Dividing two negative values results in a positive value.

$\cos^{2} (x) = \frac{3}{4}$

Step 8

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\cos (x) = \pm \sqrt{\frac{3}{4}}$

Step 9

Simplify $\pm \sqrt{\frac{3}{4}}$ .

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Step 9.1

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$\cos (x) = \pm \frac{\sqrt{3}}{\sqrt{4}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Rewrite $4$ as $2^{2}$ .

$\cos (x) = \pm \frac{\sqrt{3}}{\sqrt{2^{2}}}$

Step 9.2.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 10

The complete solution is the result of both the positive and negative portions of the solution.

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Step 10.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 10.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 10.3

The complete solution is the result of both the positive and negative portions of the solution.

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

Step 11

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

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

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

Step 12

Solve for $x$ in $\cos (x) = \frac{\sqrt{3}}{2}$ .

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Step 12.1

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

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

Step 12.2

Simplify the right side.

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Step 12.2.1

The exact value of $\arccos (\frac{\sqrt{3}}{2})$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 12.3

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.

$x = 2 π - \frac{π}{6}$

Step 12.4

Simplify $2 π - \frac{π}{6}$ .

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Step 12.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = 2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 12.4.2

Combine fractions.

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Step 12.4.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 12.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - π}{6}$

Step 12.4.3

Simplify the numerator.

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Step 12.4.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - π}{6}$

Step 12.4.3.2

Subtract $π$ from $12 π$ .

$x = \frac{11 π}{6}$

Step 12.5

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

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Step 12.5.1

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

$\frac{2 π}{| b |}$

Step 12.5.2

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

$\frac{2 π}{| 1 |}$

Step 12.5.3

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

$\frac{2 π}{1}$

Step 12.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 12.6

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

$x = \frac{π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Step 13

Solve for $x$ in $\cos (x) = - \frac{\sqrt{3}}{2}$ .

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Step 13.1

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

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

Step 13.2

Simplify the right side.

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Step 13.2.1

The exact value of $\arccos (- \frac{\sqrt{3}}{2})$ is $\frac{5 π}{6}$ .

$x = \frac{5 π}{6}$

Step 13.3

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.

$x = 2 π - \frac{5 π}{6}$

Step 13.4

Simplify $2 π - \frac{5 π}{6}$ .

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Step 13.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = 2 π \cdot \frac{6}{6} - \frac{5 π}{6}$

Step 13.4.2

Combine fractions.

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Step 13.4.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} - \frac{5 π}{6}$

Step 13.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - 5 π}{6}$

Step 13.4.3

Simplify the numerator.

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Step 13.4.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - 5 π}{6}$

Step 13.4.3.2

Subtract $5 π$ from $12 π$ .

$x = \frac{7 π}{6}$

Step 13.5

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

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Step 13.5.1

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

$\frac{2 π}{| b |}$

Step 13.5.2

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

$\frac{2 π}{| 1 |}$

Step 13.5.3

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

$\frac{2 π}{1}$

Step 13.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 13.6

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

$x = \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n$ , for any integer $n$

Step 14

List all of the solutions.

$x = \frac{π}{6} + 2 π n, \frac{11 π}{6} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n$ , for any integer $n$

Step 15

Consolidate the solutions.

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Step 15.1

Consolidate $\frac{π}{6} + 2 π n$ and $\frac{7 π}{6} + 2 π n$ to $\frac{π}{6} + π n$ .

$x = \frac{π}{6} + π n, \frac{11 π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 15.2

Consolidate $\frac{11 π}{6} + 2 π n$ and $\frac{5 π}{6} + 2 π n$ to $\frac{5 π}{6} + π n$ .

$x = \frac{π}{6} + π n, \frac{5 π}{6} + π n$ , for any integer $n$

Step 16

Find the values of $n$ that produce a value within the interval $(0, 2 π)$ .

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Step 16.1

Plug in $0$ for $n$ and simplify to see if the solution is contained in $(0, 2 π)$ .

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Step 16.1.1

Plug in $0$ for $n$ .

$\frac{π}{6} + π (0)$

Step 16.1.2

Simplify.

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Step 16.1.2.1

Multiply $π$ by $0$ .

$\frac{π}{6} + 0$

Step 16.1.2.2

Add $\frac{π}{6}$ and $0$ .

$\frac{π}{6}$

Step 16.1.3

The interval $(0, 2 π)$ contains $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 16.2

Plug in $0$ for $n$ and simplify to see if the solution is contained in $(0, 2 π)$ .

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Step 16.2.1

Plug in $0$ for $n$ .

$\frac{5 π}{6} + π (0)$

Step 16.2.2

Simplify.

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Step 16.2.2.1

Multiply $π$ by $0$ .

$\frac{5 π}{6} + 0$

Step 16.2.2.2

Add $\frac{5 π}{6}$ and $0$ .

$\frac{5 π}{6}$

Step 16.2.3

The interval $(0, 2 π)$ contains $\frac{5 π}{6}$ .

$x = \frac{π}{6}, \frac{5 π}{6}$

Step 16.3

Plug in $1$ for $n$ and simplify to see if the solution is contained in $(0, 2 π)$ .

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Step 16.3.1

Plug in $1$ for $n$ .

$\frac{π}{6} + π (1)$

Step 16.3.2

Simplify.

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Step 16.3.2.1

Multiply $π$ by $1$ .

$\frac{π}{6} + π$

Step 16.3.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{π}{6} + π \cdot \frac{6}{6}$

Step 16.3.2.3

Combine fractions.

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Step 16.3.2.3.1

Combine $π$ and $\frac{6}{6}$ .

$\frac{π}{6} + \frac{π \cdot 6}{6}$

Step 16.3.2.3.2

Combine the numerators over the common denominator.

$\frac{π + π \cdot 6}{6}$

Step 16.3.2.4

Simplify the numerator.

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Step 16.3.2.4.1

Move $6$ to the left of $π$ .

$\frac{π + 6 \cdot π}{6}$

Step 16.3.2.4.2

Add $π$ and $6 π$ .

$\frac{7 π}{6}$

Step 16.3.3

The interval $(0, 2 π)$ contains $\frac{7 π}{6}$ .

$x = \frac{π}{6}, \frac{5 π}{6}, \frac{7 π}{6}$

Step 16.4

Plug in $1$ for $n$ and simplify to see if the solution is contained in $(0, 2 π)$ .

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Step 16.4.1

Plug in $1$ for $n$ .

$\frac{5 π}{6} + π (1)$

Step 16.4.2

Simplify.

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Step 16.4.2.1

Multiply $π$ by $1$ .

$\frac{5 π}{6} + π$

Step 16.4.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{5 π}{6} + π \cdot \frac{6}{6}$

Step 16.4.2.3

Combine fractions.

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Step 16.4.2.3.1

Combine $π$ and $\frac{6}{6}$ .

$\frac{5 π}{6} + \frac{π \cdot 6}{6}$

Step 16.4.2.3.2

Combine the numerators over the common denominator.

$\frac{5 π + π \cdot 6}{6}$

Step 16.4.2.4

Simplify the numerator.

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Step 16.4.2.4.1

Move $6$ to the left of $π$ .

$\frac{5 π + 6 \cdot π}{6}$

Step 16.4.2.4.2

Add $5 π$ and $6 π$ .

$\frac{11 π}{6}$

Step 16.4.3

The interval $(0, 2 π)$ contains $\frac{11 π}{6}$ .

$x = \frac{π}{6}, \frac{5 π}{6}, \frac{7 π}{6}, \frac{11 π}{6}$