Calculus Examples

$\sum_{k = 1}^{3} {(- 1)}^{k + 4} \sin (\frac{π}{k})$

Step 1

Expand the series for each value of $k$ .

${(- 1)}^{1 + 4} \sin (\frac{π}{1}) + {(- 1)}^{2 + 4} \sin (\frac{π}{2}) + {(- 1)}^{3 + 4} \sin (\frac{π}{3})$

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Add $1$ and $4$ .

${(- 1)}^{5} \sin (\frac{π}{1}) + {(- 1)}^{2 + 4} \sin (\frac{π}{2}) + {(- 1)}^{3 + 4} \sin (\frac{π}{3})$

Step 2.1.2

Raise $- 1$ to the power of $5$ .

$- 1 \sin (\frac{π}{1}) + {(- 1)}^{2 + 4} \sin (\frac{π}{2}) + {(- 1)}^{3 + 4} \sin (\frac{π}{3})$

Step 2.1.3

Divide $π$ by $1$ .

$- 1 \sin (π) + {(- 1)}^{2 + 4} \sin (\frac{π}{2}) + {(- 1)}^{3 + 4} \sin (\frac{π}{3})$

Step 2.1.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- 1 \sin (0) + {(- 1)}^{2 + 4} \sin (\frac{π}{2}) + {(- 1)}^{3 + 4} \sin (\frac{π}{3})$

Step 2.1.5

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

$- 1 \cdot 0 + {(- 1)}^{2 + 4} \sin (\frac{π}{2}) + {(- 1)}^{3 + 4} \sin (\frac{π}{3})$

Step 2.1.6

Multiply $- 1$ by $0$ .

$0 + {(- 1)}^{2 + 4} \sin (\frac{π}{2}) + {(- 1)}^{3 + 4} \sin (\frac{π}{3})$

Step 2.1.7

Add $2$ and $4$ .

$0 + {(- 1)}^{6} \sin (\frac{π}{2}) + {(- 1)}^{3 + 4} \sin (\frac{π}{3})$

Step 2.1.8

Raise $- 1$ to the power of $6$ .

$0 + 1 \sin (\frac{π}{2}) + {(- 1)}^{3 + 4} \sin (\frac{π}{3})$

Step 2.1.9

Multiply $\sin (\frac{π}{2})$ by $1$ .

$0 + \sin (\frac{π}{2}) + {(- 1)}^{3 + 4} \sin (\frac{π}{3})$

Step 2.1.10

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

$0 + 1 + {(- 1)}^{3 + 4} \sin (\frac{π}{3})$

Step 2.1.11

Add $3$ and $4$ .

$0 + 1 + {(- 1)}^{7} \sin (\frac{π}{3})$

Step 2.1.12

Raise $- 1$ to the power of $7$ .

$0 + 1 - 1 \sin (\frac{π}{3})$

Step 2.1.13

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

$0 + 1 - 1 \frac{\sqrt{3}}{2}$

Step 2.1.14

Rewrite $- 1 \frac{\sqrt{3}}{2}$ as $- \frac{\sqrt{3}}{2}$ .

$0 + 1 - \frac{\sqrt{3}}{2}$

Step 2.2

Add $0$ and $1$ .

$1 - \frac{\sqrt{3}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$1 - \frac{\sqrt{3}}{2}$

Decimal Form:

$0.13397459 \dots$