What is the value of 1000+100sin((20pi)/(12)) ?

The value of 1000+100sin((20pi)/(12)) is 1000-50sqrt(3)

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1000+100sin((20pi)/(12))

1000 + 100 sin (\frac{20 π}{12})

1000 - 503

Decimal

913.39745 \dots

Solution steps

1000 + 100 sin (\frac{20 π}{12})

Rewrite using trig identities:

sin (\frac{20 π}{12}) = 2 sin (\frac{5 π}{6}) cos (\frac{5 π}{6})

sin (\frac{20 π}{12})

Write

sin (\frac{20 π}{12})

sin (2 \cdot \frac{5 π}{6})

= sin (2 \cdot \frac{5 π}{6})

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (\frac{5 π}{6}) cos (\frac{5 π}{6})

= 1000 + 100 \cdot 2 sin (\frac{5 π}{6}) cos (\frac{5 π}{6})

Simplify

= 1000 + 200 sin (\frac{5 π}{6}) cos (\frac{5 π}{6})

Use the following trivial identity:

sin (\frac{5 π}{6}) = \frac{1}{2}

sin (\frac{5 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}

Use the following trivial identity:

cos (\frac{5 π}{6}) = - \frac{3}{2}

cos (\frac{5 π}{6})

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{3}{2}

= 1000 + 200 \cdot \frac{1}{2} (- \frac{3}{2})

Simplify

1000 + 200 \cdot \frac{1}{2} (- \frac{3}{2}) : 1000 - 503

1000 + 200 \cdot \frac{1}{2} (- \frac{3}{2})

Remove parentheses:

(- a) = - a

= 1000 - 200 \cdot \frac{1}{2} \cdot \frac{3}{2}

200 \cdot \frac{1}{2} \cdot \frac{3}{2} = 503

200 \cdot \frac{1}{2} \cdot \frac{3}{2}

Multiply fractions:

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

= \frac{1 \cdot 3 \cdot 200}{2 \cdot 2}

Multiply the numbers:

1 \cdot 200 = 200

= \frac{200 3}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{200 3}{4}

Divide the numbers:

\frac{200}{4} = 50

= 503

= 1000 - 503

= 1000 - 503

What is the value of 1000+100sin((20pi)/(12)) ?
The value of 1000+100sin((20pi)/(12)) is 1000-50sqrt(3)