What is the value of 5sin((2pi)/3)+cos((2pi)/3) ?

The value of 5sin((2pi)/3)+cos((2pi)/3) is (5sqrt(3)-1)/2

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5sin((2pi)/3)+cos((2pi)/3)

5 sin (\frac{2 π}{3}) + cos (\frac{2 π}{3})

\frac{5 3 - 1}{2}

Decimal Notation

3.83012 \dots

Solution steps

5 sin (\frac{2 π}{3}) + cos (\frac{2 π}{3})

Use the following trivial identity:

sin (\frac{2 π}{3}) = \frac{3}{2}

sin (\frac{2 π}{3})

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{3}{2}

Use the following trivial identity:

cos (\frac{2 π}{3}) = - \frac{1}{2}

cos (\frac{2 π}{3})

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{1}{2}

= 5 \cdot \frac{3}{2} - \frac{1}{2}

Simplify

5 \cdot \frac{3}{2} - \frac{1}{2} : \frac{5 3 - 1}{2}

5 \cdot \frac{3}{2} - \frac{1}{2}

Multiply

5 \cdot \frac{3}{2} : \frac{5 3}{2}

5 \cdot \frac{3}{2}

Multiply fractions:

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

= \frac{3 \cdot 5}{2}

= \frac{5 3}{2} - \frac{1}{2}

Apply rule

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

= \frac{5 3 - 1}{2}

= \frac{5 3 - 1}{2}

\frac{tan ( 1 7 ^{\circ} ) + cot ( - 7 3 ^{\circ} )}{cot ( 1 7 ^{\circ} )}

sin (- 5 \cdot π)

cot (3 8^{\circ})

8 \cdot cos (6 0^{\circ})

\frac{sin ( 6 4 ^{\circ} )}{8} + 48 ln (2) cos (6 4^{\circ})

What is the value of 5sin((2pi)/3)+cos((2pi)/3) ?
The value of 5sin((2pi)/3)+cos((2pi)/3) is (5sqrt(3)-1)/2