What is the value of 5/3 (4pi+3)sin((4pi)/3) ?

The value of 5/3 (4pi+3)sin((4pi)/3) is -(5sqrt(3)(4pi+3))/6

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

5/3 (4pi+3)sin((4pi)/3)

Solution

\frac{5}{3} (4 π + 3) sin (\frac{4 π}{3})

Solution

- \frac{5 3 ( 4 π + 3 )}{6}

Decimal Notation

- 22.46812 \dots

Solution steps

\frac{5}{3} (4 π + 3) sin (\frac{4 π}{3})

Rewrite using trig identities:

sin (\frac{4 π}{3}) = 2 sin (\frac{2 π}{3}) cos (\frac{2 π}{3})

sin (\frac{4 π}{3})

Write

sin (\frac{4 π}{3})

sin (2 \cdot \frac{2 π}{3})

= sin (2 \cdot \frac{2 π}{3})

Use the Double Angle identity:

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

= 2 sin (\frac{2 π}{3}) cos (\frac{2 π}{3})

= \frac{5}{3} (4 π + 3) \cdot 2 sin (\frac{2 π}{3}) cos (\frac{2 π}{3})

\frac{5}{3} (4 π + 3) \cdot 2 sin (\frac{2 π}{3}) cos (\frac{2 π}{3}) = \frac{10 sin ( \frac{2 π}{3} ) cos ( \frac{2 π}{3} ) ( 4 π + 3 )}{3}

\frac{5}{3} (4 π + 3) \cdot 2 sin (\frac{2 π}{3}) cos (\frac{2 π}{3})

Multiply fractions:

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

= \frac{5 ( 4 π + 3 ) \cdot 2 sin ( \frac{2 π}{3} ) cos ( \frac{2 π}{3} )}{3}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 sin ( \frac{2 π}{3} ) cos ( \frac{2 π}{3} ) ( 4 π + 3 )}{3}

= \frac{10 sin ( \frac{2 π}{3} ) cos ( \frac{2 π}{3} ) ( 4 π + 3 )}{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}

= \frac{10 \cdot \frac{3}{2} ( - \frac{1}{2} ) ( 4 π + 3 )}{3}

Simplify

\frac{10 \cdot \frac{3}{2} ( - \frac{1}{2} ) ( 4 π + 3 )}{3} : - \frac{5 3 ( 4 π + 3 )}{6}

\frac{10 \cdot \frac{3}{2} ( - \frac{1}{2} ) ( 4 π + 3 )}{3}

Remove parentheses:

(- a) = - a

= \frac{- 10 \cdot \frac{3}{2} \cdot \frac{1}{2} ( 4 π + 3 )}{3}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{10 \cdot \frac{3}{2} \cdot \frac{1}{2} ( 4 π + 3 )}{3}

Multiply

10 \cdot \frac{3}{2} \cdot \frac{1}{2} (4 π + 3) : \frac{5 3 ( 4 π + 3 )}{2}

10 \cdot \frac{3}{2} \cdot \frac{1}{2} (4 π + 3)

Multiply fractions:

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

= \frac{3 \cdot 1 \cdot 10 ( 4 π + 3 )}{2 \cdot 2}

Multiply the numbers:

1 \cdot 10 = 10

= \frac{10 3 ( 4 π + 3 )}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{10 3 ( 4 π + 3 )}{4}

Cancel the common factor:

2

= \frac{5 3 ( 4 π + 3 )}{2}

= - \frac{\frac{5 3 ( 4 π + 3 )}{2}}{3}

Simplify

\frac{\frac{5 3 ( 4 π + 3 )}{2}}{3} : \frac{5 3 ( 4 π + 3 )}{6}

\frac{\frac{5 3 ( 4 π + 3 )}{2}}{3}

Apply the fraction rule:

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

= \frac{5 3 ( 4 π + 3 )}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{5 3 ( 4 π + 3 )}{6}

= - \frac{5 3 ( 4 π + 3 )}{6}

= - \frac{5 3 ( 4 π + 3 )}{6}

Popular Examples

4sin^2(400)

4 sin^{2} (40 0^{\circ})

cos(2400)

cos (240 0^{\circ})

tan(8/10)

tan (\frac{8}{10})

(sin(0))^2

(sin (0))^{2}

2cos^2(30)+3sin^2(30)

2 cos^{2} (3 0^{\circ}) + 3 sin^{2} (3 0^{\circ})

Frequently Asked Questions (FAQ)

What is the value of 5/3 (4pi+3)sin((4pi)/3) ?
The value of 5/3 (4pi+3)sin((4pi)/3) is -(5sqrt(3)(4pi+3))/6