Calculus Examples

$h (x) = \sin (2 x) + \cos (x)$

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 $\sin (2 x) + \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sin (2 x)] + \frac{d}{d x} [\cos (x)]$ .

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

Step 1.1.2

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

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Step 1.1.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 1.1.2.1.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

$\cos (2 x) (2 \cdot 1) + \frac{d}{d x} [\cos (x)]$

Step 1.1.2.4

Multiply $2$ by $1$ .

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

Step 1.1.2.5

Move $2$ to the left of $\cos (2 x)$ .

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

Step 1.1.3

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

$f' (x) = 2 \cos (2 x) - \sin (x)$

Step 1.2

The first derivative of $h (x)$ with respect to $x$ is $2 \cos (2 x) - \sin (x)$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $2 \cos (2 x) - \sin (x) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

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

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

Step 2.3

Simplify the left side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Multiply $2$ by $1$ .

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

Step 2.3.1.3

Multiply $- 2$ by $2$ .

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

Step 2.4

Solve the equation for $x$ .

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Step 2.4.1

Substitute $u$ for $\sin (x)$ .

$2 - 4 {(u)}^{2} - (u) = 0$

Step 2.4.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.4.3

Substitute the values $a = - 4$ , $b = - 1$ , and $c = 2$ into the quadratic formula and solve for $u$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (- 4 \cdot 2)}}{2 \cdot - 4}$

Step 2.4.4

Simplify.

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Step 2.4.4.1

Simplify the numerator.

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Step 2.4.4.1.1

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot - 4 \cdot 2}}{2 \cdot - 4}$

Step 2.4.4.1.2

Multiply $- 4 \cdot - 4 \cdot 2$ .

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Step 2.4.4.1.2.1

Multiply $- 4$ by $- 4$ .

$u = \frac{1 \pm \sqrt{1 + 16 \cdot 2}}{2 \cdot - 4}$

Step 2.4.4.1.2.2

Multiply $16$ by $2$ .

$u = \frac{1 \pm \sqrt{1 + 32}}{2 \cdot - 4}$

Step 2.4.4.1.3

Add $1$ and $32$ .

$u = \frac{1 \pm \sqrt{33}}{2 \cdot - 4}$

Step 2.4.4.2

Multiply $2$ by $- 4$ .

$u = \frac{1 \pm \sqrt{33}}{- 8}$

Step 2.4.4.3

Move the negative in front of the fraction.

$u = - \frac{1 \pm \sqrt{33}}{8}$

Step 2.4.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.4.5.1

Simplify the numerator.

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Step 2.4.5.1.1

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot - 4 \cdot 2}}{2 \cdot - 4}$

Step 2.4.5.1.2

Multiply $- 4 \cdot - 4 \cdot 2$ .

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Step 2.4.5.1.2.1

Multiply $- 4$ by $- 4$ .

$u = \frac{1 \pm \sqrt{1 + 16 \cdot 2}}{2 \cdot - 4}$

Step 2.4.5.1.2.2

Multiply $16$ by $2$ .

$u = \frac{1 \pm \sqrt{1 + 32}}{2 \cdot - 4}$

Step 2.4.5.1.3

Add $1$ and $32$ .

$u = \frac{1 \pm \sqrt{33}}{2 \cdot - 4}$

Step 2.4.5.2

Multiply $2$ by $- 4$ .

$u = \frac{1 \pm \sqrt{33}}{- 8}$

Step 2.4.5.3

Move the negative in front of the fraction.

$u = - \frac{1 \pm \sqrt{33}}{8}$

Step 2.4.5.4

Change the $\pm$ to $+$ .

$u = - \frac{1 + \sqrt{33}}{8}$

Step 2.4.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.4.6.1

Simplify the numerator.

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Step 2.4.6.1.1

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot - 4 \cdot 2}}{2 \cdot - 4}$

Step 2.4.6.1.2

Multiply $- 4 \cdot - 4 \cdot 2$ .

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Step 2.4.6.1.2.1

Multiply $- 4$ by $- 4$ .

$u = \frac{1 \pm \sqrt{1 + 16 \cdot 2}}{2 \cdot - 4}$

Step 2.4.6.1.2.2

Multiply $16$ by $2$ .

$u = \frac{1 \pm \sqrt{1 + 32}}{2 \cdot - 4}$

Step 2.4.6.1.3

Add $1$ and $32$ .

$u = \frac{1 \pm \sqrt{33}}{2 \cdot - 4}$

Step 2.4.6.2

Multiply $2$ by $- 4$ .

$u = \frac{1 \pm \sqrt{33}}{- 8}$

Step 2.4.6.3

Move the negative in front of the fraction.

$u = - \frac{1 \pm \sqrt{33}}{8}$

Step 2.4.6.4

Change the $\pm$ to $-$ .

$u = - \frac{1 - \sqrt{33}}{8}$

Step 2.4.7

The final answer is the combination of both solutions.

$u = - \frac{1 + \sqrt{33}}{8}, - \frac{1 - \sqrt{33}}{8}$

Step 2.4.8

Substitute $\sin (x)$ for $u$ .

$\sin (x) = - \frac{1 + \sqrt{33}}{8}, - \frac{1 - \sqrt{33}}{8}$

Step 2.4.9

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

$\sin (x) = - \frac{1 + \sqrt{33}}{8}$

$\sin (x) = - \frac{1 - \sqrt{33}}{8}$

Step 2.4.10

Solve for $x$ in $\sin (x) = - \frac{1 + \sqrt{33}}{8}$ .

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Step 2.4.10.1

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

$x = \arcsin (- \frac{1 + \sqrt{33}}{8})$

Step 2.4.10.2

Simplify the right side.

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Step 2.4.10.2.1

Evaluate $\arcsin (- \frac{1 + \sqrt{33}}{8})$ .

$x = - 1.00296695$

Step 2.4.10.3

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.

$x = 2 (3.14159265) + 1.00296695 + 3.14159265$

Step 2.4.10.4

Simplify the expression to find the second solution.

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Step 2.4.10.4.1

Subtract $2 π$ from $2 (3.14159265) + 1.00296695 + 3.14159265$ .

$x = 2 (3.14159265) + 1.00296695 + 3.14159265 - 2 π$

Step 2.4.10.4.2

The resulting angle of $4.1445596$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 1.00296695 + 3.14159265$ .

$x = 4.1445596$

Step 2.4.10.5

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

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Step 2.4.10.5.1

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

$\frac{2 π}{| b |}$

Step 2.4.10.5.2

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

$\frac{2 π}{| 1 |}$

Step 2.4.10.5.3

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

$\frac{2 π}{1}$

Step 2.4.10.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.4.10.6

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

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Step 2.4.10.6.1

Add $2 π$ to $- 1.00296695$ to find the positive angle.

$- 1.00296695 + 2 π$

Step 2.4.10.6.2

Subtract $1.00296695$ from $2 π$ .

$5.28021835$

Step 2.4.10.6.3

List the new angles.

$x = 5.28021835$

Step 2.4.10.7

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

$x = 4.1445596 + 2 π n, 5.28021835 + 2 π n$ , for any integer $n$

Step 2.4.11

Solve for $x$ in $\sin (x) = - \frac{1 - \sqrt{33}}{8}$ .

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Step 2.4.11.1

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

$x = \arcsin (- \frac{1 - \sqrt{33}}{8})$

Step 2.4.11.2

Simplify the right side.

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Step 2.4.11.2.1

Evaluate $\arcsin (- \frac{1 - \sqrt{33}}{8})$ .

$x = 0.63486687$

Step 2.4.11.3

$x = 2 (3.14159265) - 0.63486687 + 3.14159265$

Step 2.4.11.4

Simplify the expression to find the second solution.

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Step 2.4.11.4.1

Subtract $2 π$ from $2 (3.14159265) - 0.63486687 + 3.14159265$ .

$x = 2 (3.14159265) - 0.63486687 + 3.14159265 - 2 π$

Step 2.4.11.4.2

The resulting angle of $2.50672578$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) - 0.63486687 + 3.14159265$ .

$x = 2.50672578$

Step 2.4.11.5

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

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Step 2.4.11.5.1

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

$\frac{2 π}{| b |}$

Step 2.4.11.5.2

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

$\frac{2 π}{| 1 |}$

Step 2.4.11.5.3

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

$\frac{2 π}{1}$

Step 2.4.11.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.4.11.6

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

$x = 0.63486687 + 2 π n, 2.50672578 + 2 π n$ , for any integer $n$

Step 2.4.12

List all of the solutions.

$x = 4.1445596 + 2 π n, 5.28021835 + 2 π n, 0.63486687 + 2 π n, 2.50672578 + 2 π n$ , 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 $\sin (2 x) + \cos (x)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 4.1445596$ .

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Step 4.1.1

Substitute $4.1445596$ for $x$ .

$\sin (2 (4.1445596)) + \cos (4.1445596)$

Step 4.1.2

Multiply $2$ by $4.1445596$ .

$\sin (8.28911921) + \cos (4.1445596)$

Step 4.2

Evaluate at $x = 4.1445596 + 2 π$ .

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Step 4.2.1

Substitute $4.1445596 + 2 π$ for $x$ .

$\sin (2 (4.1445596 + 2 π)) + \cos (4.1445596 + 2 π)$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Add $4.1445596$ and $2 π$ .

$\sin (2 \cdot 10.42774491) + \cos (4.1445596 + 2 π)$

Step 4.2.2.2

Multiply $2$ by $10.42774491$ .

$\sin (20.85548982) + \cos (4.1445596 + 2 π)$

Step 4.2.2.3

Add $4.1445596$ and $2 π$ .

$\sin (20.85548982) + \cos (10.42774491)$

Step 4.3

Evaluate at $x = 4.1445596 + 4 π$ .

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Step 4.3.1

Substitute $4.1445596 + 4 π$ for $x$ .

$\sin (2 (4.1445596 + 4 π)) + \cos (4.1445596 + 4 π)$

Step 4.3.2

Simplify each term.

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Step 4.3.2.1

Add $4.1445596$ and $4 π$ .

$\sin (2 \cdot 16.71093022) + \cos (4.1445596 + 4 π)$

Step 4.3.2.2

Multiply $2$ by $16.71093022$ .

$\sin (33.42186044) + \cos (4.1445596 + 4 π)$

Step 4.3.2.3

Add $4.1445596$ and $4 π$ .

$\sin (33.42186044) + \cos (16.71093022)$

Step 4.4

Evaluate at $x = 4.1445596 + 6 π$ .

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Step 4.4.1

Substitute $4.1445596 + 6 π$ for $x$ .

$\sin (2 (4.1445596 + 6 π)) + \cos (4.1445596 + 6 π)$

Step 4.4.2

Simplify each term.

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Step 4.4.2.1

Add $4.1445596$ and $6 π$ .

$\sin (2 \cdot 22.99411552) + \cos (4.1445596 + 6 π)$

Step 4.4.2.2

Multiply $2$ by $22.99411552$ .

$\sin (45.98823105) + \cos (4.1445596 + 6 π)$

Step 4.4.2.3

Add $4.1445596$ and $6 π$ .

$\sin (45.98823105) + \cos (22.99411552)$

Step 4.5

Evaluate at $x = 4.1445596 + 8 π$ .

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Step 4.5.1

Substitute $4.1445596 + 8 π$ for $x$ .

$\sin (2 (4.1445596 + 8 π)) + \cos (4.1445596 + 8 π)$

Step 4.5.2

Simplify each term.

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Step 4.5.2.1

Add $4.1445596$ and $8 π$ .

$\sin (2 \cdot 29.27730083) + \cos (4.1445596 + 8 π)$

Step 4.5.2.2

Multiply $2$ by $29.27730083$ .

$\sin (58.55460167) + \cos (4.1445596 + 8 π)$

Step 4.5.2.3

Add $4.1445596$ and $8 π$ .

$\sin (58.55460167) + \cos (29.27730083)$

Step 4.6

Evaluate at $x = 5.28021835$ .

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Step 4.6.1

Substitute $5.28021835$ for $x$ .

$\sin (2 (5.28021835)) + \cos (5.28021835)$

Step 4.6.2

Multiply $2$ by $5.28021835$ .

$\sin (10.5604367) + \cos (5.28021835)$

Step 4.7

Evaluate at $x = 5.28021835 + 2 π$ .

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Step 4.7.1

Substitute $5.28021835 + 2 π$ for $x$ .

$\sin (2 (5.28021835 + 2 π)) + \cos (5.28021835 + 2 π)$

Step 4.7.2

Simplify each term.

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Step 4.7.2.1

Add $5.28021835$ and $2 π$ .

$\sin (2 \cdot 11.56340366) + \cos (5.28021835 + 2 π)$

Step 4.7.2.2

Multiply $2$ by $11.56340366$ .

$\sin (23.12680732) + \cos (5.28021835 + 2 π)$

Step 4.7.2.3

Add $5.28021835$ and $2 π$ .

$\sin (23.12680732) + \cos (11.56340366)$

Step 4.8

Evaluate at $x = 5.28021835 + 4 π$ .

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Step 4.8.1

Substitute $5.28021835 + 4 π$ for $x$ .

$\sin (2 (5.28021835 + 4 π)) + \cos (5.28021835 + 4 π)$

Step 4.8.2

Simplify each term.

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Step 4.8.2.1

Add $5.28021835$ and $4 π$ .

$\sin (2 \cdot 17.84658896) + \cos (5.28021835 + 4 π)$

Step 4.8.2.2

Multiply $2$ by $17.84658896$ .

$\sin (35.69317793) + \cos (5.28021835 + 4 π)$

Step 4.8.2.3

Add $5.28021835$ and $4 π$ .

$\sin (35.69317793) + \cos (17.84658896)$

Step 4.9

Evaluate at $x = 5.28021835 + 6 π$ .

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Step 4.9.1

Substitute $5.28021835 + 6 π$ for $x$ .

$\sin (2 (5.28021835 + 6 π)) + \cos (5.28021835 + 6 π)$

Step 4.9.2

Simplify each term.

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Step 4.9.2.1

Add $5.28021835$ and $6 π$ .

$\sin (2 \cdot 24.12977427) + \cos (5.28021835 + 6 π)$

Step 4.9.2.2

Multiply $2$ by $24.12977427$ .

$\sin (48.25954854) + \cos (5.28021835 + 6 π)$

Step 4.9.2.3

Add $5.28021835$ and $6 π$ .

$\sin (48.25954854) + \cos (24.12977427)$

Step 4.10

Evaluate at $x = 5.28021835 + 8 π$ .

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Step 4.10.1

Substitute $5.28021835 + 8 π$ for $x$ .

$\sin (2 (5.28021835 + 8 π)) + \cos (5.28021835 + 8 π)$

Step 4.10.2

Simplify each term.

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Step 4.10.2.1

Add $5.28021835$ and $8 π$ .

$\sin (2 \cdot 30.41295958) + \cos (5.28021835 + 8 π)$

Step 4.10.2.2

Multiply $2$ by $30.41295958$ .

$\sin (60.82591916) + \cos (5.28021835 + 8 π)$

Step 4.10.2.3

Add $5.28021835$ and $8 π$ .

$\sin (60.82591916) + \cos (30.41295958)$

Step 4.11

Evaluate at $x = 0.63486687$ .

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Step 4.11.1

Substitute $0.63486687$ for $x$ .

$\sin (2 (0.63486687)) + \cos (0.63486687)$

Step 4.11.2

Multiply $2$ by $0.63486687$ .

$\sin (1.26973374) + \cos (0.63486687)$

Step 4.12

Evaluate at $x = 0.63486687 + 2 π$ .

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Step 4.12.1

Substitute $0.63486687 + 2 π$ for $x$ .

$\sin (2 (0.63486687 + 2 π)) + \cos (0.63486687 + 2 π)$

Step 4.12.2

Simplify each term.

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Step 4.12.2.1

Add $0.63486687$ and $2 π$ .

$\sin (2 \cdot 6.91805217) + \cos (0.63486687 + 2 π)$

Step 4.12.2.2

Multiply $2$ by $6.91805217$ .

$\sin (13.83610435) + \cos (0.63486687 + 2 π)$

Step 4.12.2.3

Add $0.63486687$ and $2 π$ .

$\sin (13.83610435) + \cos (6.91805217)$

Step 4.13

Evaluate at $x = 0.63486687 + 4 π$ .

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Step 4.13.1

Substitute $0.63486687 + 4 π$ for $x$ .

$\sin (2 (0.63486687 + 4 π)) + \cos (0.63486687 + 4 π)$

Step 4.13.2

Simplify each term.

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Step 4.13.2.1

Add $0.63486687$ and $4 π$ .

$\sin (2 \cdot 13.20123748) + \cos (0.63486687 + 4 π)$

Step 4.13.2.2

Multiply $2$ by $13.20123748$ .

$\sin (26.40247497) + \cos (0.63486687 + 4 π)$

Step 4.13.2.3

Add $0.63486687$ and $4 π$ .

$\sin (26.40247497) + \cos (13.20123748)$

Step 4.14

Evaluate at $x = 0.63486687 + 6 π$ .

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Step 4.14.1

Substitute $0.63486687 + 6 π$ for $x$ .

$\sin (2 (0.63486687 + 6 π)) + \cos (0.63486687 + 6 π)$

Step 4.14.2

Simplify each term.

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Step 4.14.2.1

Add $0.63486687$ and $6 π$ .

$\sin (2 \cdot 19.48442279) + \cos (0.63486687 + 6 π)$

Step 4.14.2.2

Multiply $2$ by $19.48442279$ .

$\sin (38.96884558) + \cos (0.63486687 + 6 π)$

Step 4.14.2.3

Add $0.63486687$ and $6 π$ .

$\sin (38.96884558) + \cos (19.48442279)$

Step 4.15

Evaluate at $x = 0.63486687 + 8 π$ .

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Step 4.15.1

Substitute $0.63486687 + 8 π$ for $x$ .

$\sin (2 (0.63486687 + 8 π)) + \cos (0.63486687 + 8 π)$

Step 4.15.2

Simplify each term.

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Step 4.15.2.1

Add $0.63486687$ and $8 π$ .

$\sin (2 \cdot 25.76760809) + \cos (0.63486687 + 8 π)$

Step 4.15.2.2

Multiply $2$ by $25.76760809$ .

$\sin (51.53521619) + \cos (0.63486687 + 8 π)$

Step 4.15.2.3

Add $0.63486687$ and $8 π$ .

$\sin (51.53521619) + \cos (25.76760809)$

Step 4.16

Evaluate at $x = 2.50672578$ .

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Step 4.16.1

Substitute $2.50672578$ for $x$ .

$\sin (2 (2.50672578)) + \cos (2.50672578)$

Step 4.16.2

Multiply $2$ by $2.50672578$ .

$\sin (5.01345156) + \cos (2.50672578)$

Step 4.17

Evaluate at $x = 2.50672578 + 2 π$ .

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Step 4.17.1

Substitute $2.50672578 + 2 π$ for $x$ .

$\sin (2 (2.50672578 + 2 π)) + \cos (2.50672578 + 2 π)$

Step 4.17.2

Simplify each term.

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Step 4.17.2.1

Add $2.50672578$ and $2 π$ .

$\sin (2 \cdot 8.78991108) + \cos (2.50672578 + 2 π)$

Step 4.17.2.2

Multiply $2$ by $8.78991108$ .

$\sin (17.57982217) + \cos (2.50672578 + 2 π)$

Step 4.17.2.3

Add $2.50672578$ and $2 π$ .

$\sin (17.57982217) + \cos (8.78991108)$

Step 4.18

Evaluate at $x = 2.50672578 + 4 π$ .

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Step 4.18.1

Substitute $2.50672578 + 4 π$ for $x$ .

$\sin (2 (2.50672578 + 4 π)) + \cos (2.50672578 + 4 π)$

Step 4.18.2

Simplify each term.

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Step 4.18.2.1

Add $2.50672578$ and $4 π$ .

$\sin (2 \cdot 15.07309639) + \cos (2.50672578 + 4 π)$

Step 4.18.2.2

Multiply $2$ by $15.07309639$ .

$\sin (30.14619279) + \cos (2.50672578 + 4 π)$

Step 4.18.2.3

Add $2.50672578$ and $4 π$ .

$\sin (30.14619279) + \cos (15.07309639)$

Step 4.19

Evaluate at $x = 2.50672578 + 6 π$ .

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Step 4.19.1

Substitute $2.50672578 + 6 π$ for $x$ .

$\sin (2 (2.50672578 + 6 π)) + \cos (2.50672578 + 6 π)$

Step 4.19.2

Simplify each term.

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Step 4.19.2.1

Add $2.50672578$ and $6 π$ .

$\sin (2 \cdot 21.3562817) + \cos (2.50672578 + 6 π)$

Step 4.19.2.2

Multiply $2$ by $21.3562817$ .

$\sin (42.7125634) + \cos (2.50672578 + 6 π)$

Step 4.19.2.3

Add $2.50672578$ and $6 π$ .

$\sin (42.7125634) + \cos (21.3562817)$

Step 4.20

Evaluate at $x = 2.50672578 + 8 π$ .

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Step 4.20.1

Substitute $2.50672578 + 8 π$ for $x$ .

$\sin (2 (2.50672578 + 8 π)) + \cos (2.50672578 + 8 π)$

Step 4.20.2

Simplify each term.

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Step 4.20.2.1

Add $2.50672578$ and $8 π$ .

$\sin (2 \cdot 27.63946701) + \cos (2.50672578 + 8 π)$

Step 4.20.2.2

Multiply $2$ by $27.63946701$ .

$\sin (55.27893402) + \cos (2.50672578 + 8 π)$

Step 4.20.2.3

Add $2.50672578$ and $8 π$ .

$\sin (55.27893402) + \cos (27.63946701)$

Step 4.21

List all of the points.

$(4.1445596, \sin (8.28911921) + \cos (4.1445596)), (4.1445596 + 2 π, \sin (20.85548982) + \cos (10.42774491)), (4.1445596 + 4 π, \sin (33.42186044) + \cos (16.71093022)), (4.1445596 + 6 π, \sin (45.98823105) + \cos (22.99411552)), (4.1445596 + 8 π, \sin (58.55460167) + \cos (29.27730083)), (5.28021835, \sin (10.5604367) + \cos (5.28021835)), (5.28021835 + 2 π, \sin (23.12680732) + \cos (11.56340366)), (5.28021835 + 4 π, \sin (35.69317793) + \cos (17.84658896)), (5.28021835 + 6 π, \sin (48.25954854) + \cos (24.12977427)), (5.28021835 + 8 π, \sin (60.82591916) + \cos (30.41295958)), (0.63486687, \sin (1.26973374) + \cos (0.63486687)), (0.63486687 + 2 π, \sin (13.83610435) + \cos (6.91805217)), (0.63486687 + 4 π, \sin (26.40247497) + \cos (13.20123748)), (0.63486687 + 6 π, \sin (38.96884558) + \cos (19.48442279)), (0.63486687 + 8 π, \sin (51.53521619) + \cos (25.76760809)), (2.50672578, \sin (5.01345156) + \cos (2.50672578)), (2.50672578 + 2 π, \sin (17.57982217) + \cos (8.78991108)), (2.50672578 + 4 π, \sin (30.14619279) + \cos (15.07309639)), (2.50672578 + 6 π, \sin (42.7125634) + \cos (21.3562817)), (2.50672578 + 8 π, \sin (55.27893402) + \cos (27.63946701))$ , for any integer $n$

Step 5