Calculus Examples

$\sum_{k = 1}^{4} 2 \sin (\frac{π}{k})$

Step 1

Expand the series for each value of $k$ .

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

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Divide $π$ by $1$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Multiply $2$ by $0$ .

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

Step 2.1.5

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

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

Step 2.1.6

Multiply $2$ by $1$ .

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

Step 2.1.7

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

$0 + 2 + 2 \frac{\sqrt{3}}{2} + 2 \sin (\frac{π}{4})$

Step 2.1.8

Cancel the common factor of $2$ .

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Step 2.1.8.1

Cancel the common factor.

$0 + 2 + 2 \frac{\sqrt{3}}{2} + 2 \sin (\frac{π}{4})$

Step 2.1.8.2

Rewrite the expression.

$0 + 2 + \sqrt{3} + 2 \sin (\frac{π}{4})$

Step 2.1.9

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 2.1.10

Cancel the common factor of $2$ .

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Step 2.1.10.1

Cancel the common factor.

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

Step 2.1.10.2

Rewrite the expression.

$0 + 2 + \sqrt{3} + \sqrt{2}$

Step 2.2

Add $0$ and $2$ .

$2 + \sqrt{3} + \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$2 + \sqrt{3} + \sqrt{2}$

Decimal Form:

$5.14626436 \dots$