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العربية

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

الحلّ

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

- \frac{5 3 ( 4 π + 3 )}{6}

عشري

- 22.46812 \dots

خطوات الحلّ

\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})

sin (2 \cdot \frac{2 π}{3})

كـ

sin (\frac{4 π}{3})

أكتب

= sin (2 \cdot \frac{2 π}{3})

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

:فعّل متطابقة الزاوية المضاعفة

= 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})

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

:اضرب كسور

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

5 \cdot 2 = 10 :

اضرب الأعداد

= \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}

استخدم المتطابقة الأساسيّة التالية:

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}

استخدم المتطابقة الأساسيّة التالية:

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}

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

بسّط:

- \frac{5 3 ( 4 π + 3 )}{6}

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

(- a) = - a

:احذف الأقواس

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

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

: استخدم ميزات الكسور التالية

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

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

اضرب بـ:

\frac{5 3 ( 4 π + 3 )}{2}

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

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

:اضرب كسور

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

1 \cdot 10 = 10 :

اضرب الأعداد

= \frac{10 3 ( 4 π + 3 )}{2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{10 3 ( 4 π + 3 )}{4}

2 :

إلغ العوامل المشتركة

= \frac{5 3 ( 4 π + 3 )}{2}

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

\frac{\frac{5 3 ( 4 π + 3 )}{2}}{3}

بسّط:

\frac{5 3 ( 4 π + 3 )}{6}

\frac{\frac{5 3 ( 4 π + 3 )}{2}}{3}

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

: استخدم ميزات الكسور التالية

= \frac{5 3 ( 4 π + 3 )}{2 \cdot 3}

2 \cdot 3 = 6 :

اضرب الأعداد

= \frac{5 3 ( 4 π + 3 )}{6}

= - \frac{5 3 ( 4 π + 3 )}{6}

= - \frac{5 3 ( 4 π + 3 )}{6}

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

cos (240 0^{\circ})

(sin (0))^{2}

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

\frac{tan ( 31 5 ^{\circ} ) - cos ( 78 0 ^{\circ} )}{tan ( - 13 5 ^{\circ} ) - sin ( 21 0 ^{\circ} )}