Calculus Examples

$2 \cos (4 x) + 4 x$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $2 \cos (4 x) + 4 x$ with respect to $x$ is $\frac{d}{d x} [2 \cos (4 x)] + \frac{d}{d x} [4 x]$ .

$\frac{d}{d x} [2 \cos (4 x)] + \frac{d}{d x} [4 x]$

Step 1.1.2

Evaluate $\frac{d}{d x} [2 \cos (4 x)]$ .

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Step 1.1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \cos (4 x)$ with respect to $x$ is $2 \frac{d}{d x} [\cos (4 x)]$ .

$2 \frac{d}{d x} [\cos (4 x)] + \frac{d}{d x} [4 x]$

Step 1.1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 4 x$ .

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Step 1.1.2.2.1

To apply the Chain Rule, set $u$ as $4 x$ .

$2 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [4 x]) + \frac{d}{d x} [4 x]$

Step 1.1.2.2.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$2 (- \sin (u) \frac{d}{d x} [4 x]) + \frac{d}{d x} [4 x]$

Step 1.1.2.2.3

Replace all occurrences of $u$ with $4 x$ .

$2 (- \sin (4 x) \frac{d}{d x} [4 x]) + \frac{d}{d x} [4 x]$

Step 1.1.2.3

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$2 (- \sin (4 x) (4 \frac{d}{d x} [x])) + \frac{d}{d x} [4 x]$

Step 1.1.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 (- \sin (4 x) (4 \cdot 1)) + \frac{d}{d x} [4 x]$

Step 1.1.2.5

Multiply $4$ by $1$ .

$2 (- \sin (4 x) \cdot 4) + \frac{d}{d x} [4 x]$

Step 1.1.2.6

Multiply $4$ by $- 1$ .

$2 (- 4 \sin (4 x)) + \frac{d}{d x} [4 x]$

Step 1.1.2.7

Multiply $- 4$ by $2$ .

$- 8 \sin (4 x) + \frac{d}{d x} [4 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$- 8 \sin (4 x) + 4 \frac{d}{d x} [x]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 8 \sin (4 x) + 4 \cdot 1$

Step 1.1.3.3

Multiply $4$ by $1$ .

$- 8 \sin (4 x) + 4$

Step 1.1.4

Reorder terms.

$f' (x) = 4 - 8 \sin (4 x)$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $4 - 8 \sin (4 x)$ .

$4 - 8 \sin (4 x)$

Step 2

Set the first derivative equal to $0$ then solve the equation $4 - 8 \sin (4 x) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$4 - 8 \sin (4 x) = 0$

Step 2.2

Subtract $4$ from both sides of the equation.

$- 8 \sin (4 x) = - 4$

Step 2.3

Divide each term in $- 8 \sin (4 x) = - 4$ by $- 8$ and simplify.

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Step 2.3.1

Divide each term in $- 8 \sin (4 x) = - 4$ by $- 8$ .

$\frac{- 8 \sin (4 x)}{- 8} = \frac{- 4}{- 8}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 8$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 8 \sin (4 x)}{- 8} = \frac{- 4}{- 8}$

Step 2.3.2.1.2

Divide $\sin (4 x)$ by $1$ .

$\sin (4 x) = \frac{- 4}{- 8}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $- 4$ and $- 8$ .

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Step 2.3.3.1.1

Factor $- 4$ out of $- 4$ .

$\sin (4 x) = \frac{- 4 (1)}{- 8}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor $- 4$ out of $- 8$ .

$\sin (4 x) = \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 2.3.3.1.2.2

Cancel the common factor.

$\sin (4 x) = \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 2.3.3.1.2.3

Rewrite the expression.

$\sin (4 x) = \frac{1}{2}$

Step 2.4

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

$4 x = \arcsin (\frac{1}{2})$

Step 2.5

Simplify the right side.

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Step 2.5.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$4 x = \frac{π}{6}$

Step 2.6

Divide each term in $4 x = \frac{π}{6}$ by $4$ and simplify.

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Step 2.6.1

Divide each term in $4 x = \frac{π}{6}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{π}{6}}{4}$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Cancel the common factor of $4$ .

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Step 2.6.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{π}{6}}{4}$

Step 2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{6}}{4}$

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{6} \cdot \frac{1}{4}$

Step 2.6.3.2

Multiply $\frac{π}{6} \cdot \frac{1}{4}$ .

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Step 2.6.3.2.1

Multiply $\frac{π}{6}$ by $\frac{1}{4}$ .

$x = \frac{π}{6 \cdot 4}$

Step 2.6.3.2.2

Multiply $6$ by $4$ .

$x = \frac{π}{24}$

Step 2.7

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.

$4 x = π - \frac{π}{6}$

Step 2.8

Solve for $x$ .

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Step 2.8.1

Simplify.

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Step 2.8.1.1

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

$4 x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 2.8.1.2

Combine $π$ and $\frac{6}{6}$ .

$4 x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 2.8.1.3

Combine the numerators over the common denominator.

$4 x = \frac{π \cdot 6 - π}{6}$

Step 2.8.1.4

Subtract $π$ from $π \cdot 6$ .

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Step 2.8.1.4.1

Reorder $π$ and $6$ .

$4 x = \frac{6 \cdot π - π}{6}$

Step 2.8.1.4.2

Subtract $π$ from $6 \cdot π$ .

$4 x = \frac{5 \cdot π}{6}$

Step 2.8.2

Divide each term in $4 x = \frac{5 \cdot π}{6}$ by $4$ and simplify.

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Step 2.8.2.1

Divide each term in $4 x = \frac{5 \cdot π}{6}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{5 \cdot π}{6}}{4}$

Step 2.8.2.2

Simplify the left side.

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Step 2.8.2.2.1

Cancel the common factor of $4$ .

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Step 2.8.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{5 \cdot π}{6}}{4}$

Step 2.8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 \cdot π}{6}}{4}$

Step 2.8.2.3

Simplify the right side.

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Step 2.8.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 \cdot π}{6} \cdot \frac{1}{4}$

Step 2.8.2.3.2

Multiply $\frac{5 π}{6} \cdot \frac{1}{4}$ .

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Step 2.8.2.3.2.1

Multiply $\frac{5 π}{6}$ by $\frac{1}{4}$ .

$x = \frac{5 π}{6 \cdot 4}$

Step 2.8.2.3.2.2

Multiply $6$ by $4$ .

$x = \frac{5 π}{24}$

Step 2.9

Find the period of $\sin (4 x)$ .

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Step 2.9.1

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

$\frac{2 π}{| b |}$

Step 2.9.2

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 2.9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Step 2.9.4

Cancel the common factor of $2$ and $4$ .

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Step 2.9.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 2.9.4.2

Cancel the common factors.

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Step 2.9.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 2.9.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 2.9.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 2.10

The period of the $\sin (4 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{π}{24} + \frac{π n}{2}, \frac{5 π}{24} + \frac{π n}{2}$ , for any integer $n$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $2 \cos (4 x) + 4 x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{π}{24}$ .

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Step 4.1.1

Substitute $\frac{π}{24}$ for $x$ .

$2 \cos (4 (\frac{π}{24})) + 4 (\frac{π}{24})$

Step 4.1.2

Simplify each term.

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Step 4.1.2.1

Cancel the common factor of $4$ .

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Step 4.1.2.1.1

Factor $4$ out of $24$ .

$2 \cos (4 \frac{π}{4 (6)}) + 4 (\frac{π}{24})$

Step 4.1.2.1.2

Cancel the common factor.

$2 \cos (4 \frac{π}{4 \cdot 6}) + 4 (\frac{π}{24})$

Step 4.1.2.1.3

Rewrite the expression.

$2 \cos (\frac{π}{6}) + 4 (\frac{π}{24})$

Step 4.1.2.2

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

$2 \frac{\sqrt{3}}{2} + 4 (\frac{π}{24})$

Step 4.1.2.3

Cancel the common factor of $2$ .

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Step 4.1.2.3.1

Cancel the common factor.

$2 \frac{\sqrt{3}}{2} + 4 (\frac{π}{24})$

Step 4.1.2.3.2

Rewrite the expression.

$\sqrt{3} + 4 (\frac{π}{24})$

Step 4.1.2.4

Cancel the common factor of $4$ .

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Step 4.1.2.4.1

Factor $4$ out of $24$ .

$\sqrt{3} + 4 \frac{π}{4 (6)}$

Step 4.1.2.4.2

Cancel the common factor.

$\sqrt{3} + 4 \frac{π}{4 \cdot 6}$

Step 4.1.2.4.3

Rewrite the expression.

$\sqrt{3} + \frac{π}{6}$

Step 4.2

Evaluate at $x = \frac{13 π}{24}$ .

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Step 4.2.1

Substitute $\frac{13 π}{24}$ for $x$ .

$2 \cos (4 (\frac{13 π}{24})) + 4 (\frac{13 π}{24})$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Factor $4$ out of $24$ .

$2 \cos (4 \frac{13 π}{4 (6)}) + 4 (\frac{13 π}{24})$

Step 4.2.2.1.2

Cancel the common factor.

$2 \cos (4 \frac{13 π}{4 \cdot 6}) + 4 (\frac{13 π}{24})$

Step 4.2.2.1.3

Rewrite the expression.

$2 \cos (\frac{13 π}{6}) + 4 (\frac{13 π}{24})$

Step 4.2.2.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$2 \cos (\frac{π}{6}) + 4 (\frac{13 π}{24})$

Step 4.2.2.3

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

$2 \frac{\sqrt{3}}{2} + 4 (\frac{13 π}{24})$

Step 4.2.2.4

Cancel the common factor of $2$ .

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Step 4.2.2.4.1

Cancel the common factor.

$2 \frac{\sqrt{3}}{2} + 4 (\frac{13 π}{24})$

Step 4.2.2.4.2

Rewrite the expression.

$\sqrt{3} + 4 (\frac{13 π}{24})$

Step 4.2.2.5

Cancel the common factor of $4$ .

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Step 4.2.2.5.1

Factor $4$ out of $24$ .

$\sqrt{3} + 4 \frac{13 π}{4 (6)}$

Step 4.2.2.5.2

Cancel the common factor.

$\sqrt{3} + 4 \frac{13 π}{4 \cdot 6}$

Step 4.2.2.5.3

Rewrite the expression.

$\sqrt{3} + \frac{13 π}{6}$

Step 4.3

Evaluate at $x = \frac{25 π}{24}$ .

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Step 4.3.1

Substitute $\frac{25 π}{24}$ for $x$ .

$2 \cos (4 (\frac{25 π}{24})) + 4 (\frac{25 π}{24})$

Step 4.3.2

Simplify each term.

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Step 4.3.2.1

Cancel the common factor of $4$ .

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Step 4.3.2.1.1

Factor $4$ out of $24$ .

$2 \cos (4 \frac{25 π}{4 (6)}) + 4 (\frac{25 π}{24})$

Step 4.3.2.1.2

Cancel the common factor.

$2 \cos (4 \frac{25 π}{4 \cdot 6}) + 4 (\frac{25 π}{24})$

Step 4.3.2.1.3

Rewrite the expression.

$2 \cos (\frac{25 π}{6}) + 4 (\frac{25 π}{24})$

Step 4.3.2.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$2 \cos (\frac{π}{6}) + 4 (\frac{25 π}{24})$

Step 4.3.2.3

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

$2 \frac{\sqrt{3}}{2} + 4 (\frac{25 π}{24})$

Step 4.3.2.4

Cancel the common factor of $2$ .

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Step 4.3.2.4.1

Cancel the common factor.

$2 \frac{\sqrt{3}}{2} + 4 (\frac{25 π}{24})$

Step 4.3.2.4.2

Rewrite the expression.

$\sqrt{3} + 4 (\frac{25 π}{24})$

Step 4.3.2.5

Cancel the common factor of $4$ .

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Step 4.3.2.5.1

Factor $4$ out of $24$ .

$\sqrt{3} + 4 \frac{25 π}{4 (6)}$

Step 4.3.2.5.2

Cancel the common factor.

$\sqrt{3} + 4 \frac{25 π}{4 \cdot 6}$

Step 4.3.2.5.3

Rewrite the expression.

$\sqrt{3} + \frac{25 π}{6}$

Step 4.4

Evaluate at $x = \frac{37 π}{24}$ .

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Step 4.4.1

Substitute $\frac{37 π}{24}$ for $x$ .

$2 \cos (4 (\frac{37 π}{24})) + 4 (\frac{37 π}{24})$

Step 4.4.2

Simplify each term.

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Step 4.4.2.1

Cancel the common factor of $4$ .

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Step 4.4.2.1.1

Factor $4$ out of $24$ .

$2 \cos (4 \frac{37 π}{4 (6)}) + 4 (\frac{37 π}{24})$

Step 4.4.2.1.2

Cancel the common factor.

$2 \cos (4 \frac{37 π}{4 \cdot 6}) + 4 (\frac{37 π}{24})$

Step 4.4.2.1.3

Rewrite the expression.

$2 \cos (\frac{37 π}{6}) + 4 (\frac{37 π}{24})$

Step 4.4.2.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$2 \cos (\frac{π}{6}) + 4 (\frac{37 π}{24})$

Step 4.4.2.3

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

$2 \frac{\sqrt{3}}{2} + 4 (\frac{37 π}{24})$

Step 4.4.2.4

Cancel the common factor of $2$ .

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Step 4.4.2.4.1

Cancel the common factor.

$2 \frac{\sqrt{3}}{2} + 4 (\frac{37 π}{24})$

Step 4.4.2.4.2

Rewrite the expression.

$\sqrt{3} + 4 (\frac{37 π}{24})$

Step 4.4.2.5

Cancel the common factor of $4$ .

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Step 4.4.2.5.1

Factor $4$ out of $24$ .

$\sqrt{3} + 4 \frac{37 π}{4 (6)}$

Step 4.4.2.5.2

Cancel the common factor.

$\sqrt{3} + 4 \frac{37 π}{4 \cdot 6}$

Step 4.4.2.5.3

Rewrite the expression.

$\sqrt{3} + \frac{37 π}{6}$

Step 4.5

Evaluate at $x = \frac{49 π}{24}$ .

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Step 4.5.1

Substitute $\frac{49 π}{24}$ for $x$ .

$2 \cos (4 (\frac{49 π}{24})) + 4 (\frac{49 π}{24})$

Step 4.5.2

Simplify each term.

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Step 4.5.2.1

Cancel the common factor of $4$ .

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Step 4.5.2.1.1

Factor $4$ out of $24$ .

$2 \cos (4 \frac{49 π}{4 (6)}) + 4 (\frac{49 π}{24})$

Step 4.5.2.1.2

Cancel the common factor.

$2 \cos (4 \frac{49 π}{4 \cdot 6}) + 4 (\frac{49 π}{24})$

Step 4.5.2.1.3

Rewrite the expression.

$2 \cos (\frac{49 π}{6}) + 4 (\frac{49 π}{24})$

Step 4.5.2.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$2 \cos (\frac{π}{6}) + 4 (\frac{49 π}{24})$

Step 4.5.2.3

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

$2 \frac{\sqrt{3}}{2} + 4 (\frac{49 π}{24})$

Step 4.5.2.4

Cancel the common factor of $2$ .

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Step 4.5.2.4.1

Cancel the common factor.

$2 \frac{\sqrt{3}}{2} + 4 (\frac{49 π}{24})$

Step 4.5.2.4.2

Rewrite the expression.

$\sqrt{3} + 4 (\frac{49 π}{24})$

Step 4.5.2.5

Cancel the common factor of $4$ .

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Step 4.5.2.5.1

Factor $4$ out of $24$ .

$\sqrt{3} + 4 \frac{49 π}{4 (6)}$

Step 4.5.2.5.2

Cancel the common factor.

$\sqrt{3} + 4 \frac{49 π}{4 \cdot 6}$

Step 4.5.2.5.3

Rewrite the expression.

$\sqrt{3} + \frac{49 π}{6}$

Step 4.6

Evaluate at $x = \frac{5 π}{24}$ .

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Step 4.6.1

Substitute $\frac{5 π}{24}$ for $x$ .

$2 \cos (4 (\frac{5 π}{24})) + 4 (\frac{5 π}{24})$

Step 4.6.2

Simplify each term.

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Step 4.6.2.1

Cancel the common factor of $4$ .

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Step 4.6.2.1.1

Factor $4$ out of $24$ .

$2 \cos (4 \frac{5 π}{4 (6)}) + 4 (\frac{5 π}{24})$

Step 4.6.2.1.2

Cancel the common factor.

$2 \cos (4 \frac{5 π}{4 \cdot 6}) + 4 (\frac{5 π}{24})$

Step 4.6.2.1.3

Rewrite the expression.

$2 \cos (\frac{5 π}{6}) + 4 (\frac{5 π}{24})$

Step 4.6.2.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$2 (- \cos (\frac{π}{6})) + 4 (\frac{5 π}{24})$

Step 4.6.2.3

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

$2 (- \frac{\sqrt{3}}{2}) + 4 (\frac{5 π}{24})$

Step 4.6.2.4

Cancel the common factor of $2$ .

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Step 4.6.2.4.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$2 \frac{- \sqrt{3}}{2} + 4 (\frac{5 π}{24})$

Step 4.6.2.4.2

Cancel the common factor.

$2 \frac{- \sqrt{3}}{2} + 4 (\frac{5 π}{24})$

Step 4.6.2.4.3

Rewrite the expression.

$- \sqrt{3} + 4 (\frac{5 π}{24})$

Step 4.6.2.5

Cancel the common factor of $4$ .

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Step 4.6.2.5.1

Factor $4$ out of $24$ .

$- \sqrt{3} + 4 \frac{5 π}{4 (6)}$

Step 4.6.2.5.2

Cancel the common factor.

$- \sqrt{3} + 4 \frac{5 π}{4 \cdot 6}$

Step 4.6.2.5.3

Rewrite the expression.

$- \sqrt{3} + \frac{5 π}{6}$

Step 4.7

Evaluate at $x = \frac{17 π}{24}$ .

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Step 4.7.1

Substitute $\frac{17 π}{24}$ for $x$ .

$2 \cos (4 (\frac{17 π}{24})) + 4 (\frac{17 π}{24})$

Step 4.7.2

Simplify each term.

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Step 4.7.2.1

Cancel the common factor of $4$ .

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Step 4.7.2.1.1

Factor $4$ out of $24$ .

$2 \cos (4 \frac{17 π}{4 (6)}) + 4 (\frac{17 π}{24})$

Step 4.7.2.1.2

Cancel the common factor.

$2 \cos (4 \frac{17 π}{4 \cdot 6}) + 4 (\frac{17 π}{24})$

Step 4.7.2.1.3

Rewrite the expression.

$2 \cos (\frac{17 π}{6}) + 4 (\frac{17 π}{24})$

Step 4.7.2.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$2 \cos (\frac{5 π}{6}) + 4 (\frac{17 π}{24})$

Step 4.7.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$2 (- \cos (\frac{π}{6})) + 4 (\frac{17 π}{24})$

Step 4.7.2.4

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

$2 (- \frac{\sqrt{3}}{2}) + 4 (\frac{17 π}{24})$

Step 4.7.2.5

Cancel the common factor of $2$ .

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Step 4.7.2.5.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$2 \frac{- \sqrt{3}}{2} + 4 (\frac{17 π}{24})$

Step 4.7.2.5.2

Cancel the common factor.

$2 \frac{- \sqrt{3}}{2} + 4 (\frac{17 π}{24})$

Step 4.7.2.5.3

Rewrite the expression.

$- \sqrt{3} + 4 (\frac{17 π}{24})$

Step 4.7.2.6

Cancel the common factor of $4$ .

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Step 4.7.2.6.1

Factor $4$ out of $24$ .

$- \sqrt{3} + 4 \frac{17 π}{4 (6)}$

Step 4.7.2.6.2

Cancel the common factor.

$- \sqrt{3} + 4 \frac{17 π}{4 \cdot 6}$

Step 4.7.2.6.3

Rewrite the expression.

$- \sqrt{3} + \frac{17 π}{6}$

Step 4.8

Evaluate at $x = \frac{29 π}{24}$ .

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Step 4.8.1

Substitute $\frac{29 π}{24}$ for $x$ .

$2 \cos (4 (\frac{29 π}{24})) + 4 (\frac{29 π}{24})$

Step 4.8.2

Simplify each term.

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Step 4.8.2.1

Cancel the common factor of $4$ .

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Step 4.8.2.1.1

Factor $4$ out of $24$ .

$2 \cos (4 \frac{29 π}{4 (6)}) + 4 (\frac{29 π}{24})$

Step 4.8.2.1.2

Cancel the common factor.

$2 \cos (4 \frac{29 π}{4 \cdot 6}) + 4 (\frac{29 π}{24})$

Step 4.8.2.1.3

Rewrite the expression.

$2 \cos (\frac{29 π}{6}) + 4 (\frac{29 π}{24})$

Step 4.8.2.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$2 \cos (\frac{5 π}{6}) + 4 (\frac{29 π}{24})$

Step 4.8.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$2 (- \cos (\frac{π}{6})) + 4 (\frac{29 π}{24})$

Step 4.8.2.4

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

$2 (- \frac{\sqrt{3}}{2}) + 4 (\frac{29 π}{24})$

Step 4.8.2.5

Cancel the common factor of $2$ .

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Step 4.8.2.5.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$2 \frac{- \sqrt{3}}{2} + 4 (\frac{29 π}{24})$

Step 4.8.2.5.2

Cancel the common factor.

$2 \frac{- \sqrt{3}}{2} + 4 (\frac{29 π}{24})$

Step 4.8.2.5.3

Rewrite the expression.

$- \sqrt{3} + 4 (\frac{29 π}{24})$

Step 4.8.2.6

Cancel the common factor of $4$ .

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Step 4.8.2.6.1

Factor $4$ out of $24$ .

$- \sqrt{3} + 4 \frac{29 π}{4 (6)}$

Step 4.8.2.6.2

Cancel the common factor.

$- \sqrt{3} + 4 \frac{29 π}{4 \cdot 6}$

Step 4.8.2.6.3

Rewrite the expression.

$- \sqrt{3} + \frac{29 π}{6}$

Step 4.9

Evaluate at $x = \frac{41 π}{24}$ .

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Step 4.9.1

Substitute $\frac{41 π}{24}$ for $x$ .

$2 \cos (4 (\frac{41 π}{24})) + 4 (\frac{41 π}{24})$

Step 4.9.2

Simplify each term.

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Step 4.9.2.1

Cancel the common factor of $4$ .

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Step 4.9.2.1.1

Factor $4$ out of $24$ .

$2 \cos (4 \frac{41 π}{4 (6)}) + 4 (\frac{41 π}{24})$

Step 4.9.2.1.2

Cancel the common factor.

$2 \cos (4 \frac{41 π}{4 \cdot 6}) + 4 (\frac{41 π}{24})$

Step 4.9.2.1.3

Rewrite the expression.

$2 \cos (\frac{41 π}{6}) + 4 (\frac{41 π}{24})$

Step 4.9.2.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$2 \cos (\frac{5 π}{6}) + 4 (\frac{41 π}{24})$

Step 4.9.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$2 (- \cos (\frac{π}{6})) + 4 (\frac{41 π}{24})$

Step 4.9.2.4

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

$2 (- \frac{\sqrt{3}}{2}) + 4 (\frac{41 π}{24})$

Step 4.9.2.5

Cancel the common factor of $2$ .

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Step 4.9.2.5.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$2 \frac{- \sqrt{3}}{2} + 4 (\frac{41 π}{24})$

Step 4.9.2.5.2

Cancel the common factor.

$2 \frac{- \sqrt{3}}{2} + 4 (\frac{41 π}{24})$

Step 4.9.2.5.3

Rewrite the expression.

$- \sqrt{3} + 4 (\frac{41 π}{24})$

Step 4.9.2.6

Cancel the common factor of $4$ .

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Step 4.9.2.6.1

Factor $4$ out of $24$ .

$- \sqrt{3} + 4 \frac{41 π}{4 (6)}$

Step 4.9.2.6.2

Cancel the common factor.

$- \sqrt{3} + 4 \frac{41 π}{4 \cdot 6}$

Step 4.9.2.6.3

Rewrite the expression.

$- \sqrt{3} + \frac{41 π}{6}$

Step 4.10

Evaluate at $x = \frac{53 π}{24}$ .

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Step 4.10.1

Substitute $\frac{53 π}{24}$ for $x$ .

$2 \cos (4 (\frac{53 π}{24})) + 4 (\frac{53 π}{24})$

Step 4.10.2

Simplify each term.

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Step 4.10.2.1

Cancel the common factor of $4$ .

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Step 4.10.2.1.1

Factor $4$ out of $24$ .

$2 \cos (4 \frac{53 π}{4 (6)}) + 4 (\frac{53 π}{24})$

Step 4.10.2.1.2

Cancel the common factor.

$2 \cos (4 \frac{53 π}{4 \cdot 6}) + 4 (\frac{53 π}{24})$

Step 4.10.2.1.3

Rewrite the expression.

$2 \cos (\frac{53 π}{6}) + 4 (\frac{53 π}{24})$

Step 4.10.2.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$2 \cos (\frac{5 π}{6}) + 4 (\frac{53 π}{24})$

Step 4.10.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$2 (- \cos (\frac{π}{6})) + 4 (\frac{53 π}{24})$

Step 4.10.2.4

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

$2 (- \frac{\sqrt{3}}{2}) + 4 (\frac{53 π}{24})$

Step 4.10.2.5

Cancel the common factor of $2$ .

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Step 4.10.2.5.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$2 \frac{- \sqrt{3}}{2} + 4 (\frac{53 π}{24})$

Step 4.10.2.5.2

Cancel the common factor.

$2 \frac{- \sqrt{3}}{2} + 4 (\frac{53 π}{24})$

Step 4.10.2.5.3

Rewrite the expression.

$- \sqrt{3} + 4 (\frac{53 π}{24})$

Step 4.10.2.6

Cancel the common factor of $4$ .

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Step 4.10.2.6.1

Factor $4$ out of $24$ .

$- \sqrt{3} + 4 \frac{53 π}{4 (6)}$

Step 4.10.2.6.2

Cancel the common factor.

$- \sqrt{3} + 4 \frac{53 π}{4 \cdot 6}$

Step 4.10.2.6.3

Rewrite the expression.

$- \sqrt{3} + \frac{53 π}{6}$

Step 4.11

List all of the points.

$(\frac{π}{24}, \sqrt{3} + \frac{π}{6}), (\frac{13 π}{24}, \sqrt{3} + \frac{13 π}{6}), (\frac{25 π}{24}, \sqrt{3} + \frac{25 π}{6}), (\frac{37 π}{24}, \sqrt{3} + \frac{37 π}{6}), (\frac{49 π}{24}, \sqrt{3} + \frac{49 π}{6}), (\frac{5 π}{24}, - \sqrt{3} + \frac{5 π}{6}), (\frac{17 π}{24}, - \sqrt{3} + \frac{17 π}{6}), (\frac{29 π}{24}, - \sqrt{3} + \frac{29 π}{6}), (\frac{41 π}{24}, - \sqrt{3} + \frac{41 π}{6}), (\frac{53 π}{24}, - \sqrt{3} + \frac{53 π}{6})$

Step 5