What is the value of e^{2sin(pi/6)} ?

The value of e^{2sin(pi/6)} is e

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e^{2sin(pi/6)}

e^{2 s i n (\frac{π}{6})}

e

Decimal

2.71828 \dots

Solution steps

e^{2 s i n (\frac{π}{6})}

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

sin (\frac{π}{6})

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= e^{2 \cdot \frac{1}{2}}

Simplify

e^{2 \cdot \frac{1}{2}} : e

e^{2 \cdot \frac{1}{2}}

Multiply

2 \cdot \frac{1}{2} : 1

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= e^{1}

Apply rule

a^{1} = a

= e

= e

cot (\frac{13 π}{12})

6 cos (10 5^{\circ}) + 6 cos (1 5^{\circ})

6 cos (13 5^{\circ}) - 6 cos (4 5^{\circ})

cos (\frac{π}{6}) - sin (\frac{3 π}{4})

- 1 - arctan (- 1)