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شائع علم المثلثات >

prove (cos^2(2θ))/2 =(1+cos(4θ))/8

الحلّ

prove

\frac{cos ^{2} ( 2 θ )}{2} = \frac{1 + cos ( 4 θ )}{8}

الحلّ

خطأ

خطوات الحلّ

\frac{cos ^{2} ( 2 θ )}{2} = \frac{1 + cos ( 4 θ )}{8}

لا يتساوى الطرفان

θ = 0

عوّض

, \frac{cos ^{2} ( 2 θ )}{2} = \frac{1 + cos ( 4 θ )}{8}

في

\frac{cos ^{2} ( 2 \cdot 0 )}{2} = \frac{1}{2} (Decimal : 0.5)

\frac{cos ^{2} ( 2 \cdot 0 )}{2}

بسّط

= \frac{cos ^{2} ( 0 )}{2}

استخدم المتطابقة الأساسيّة التالية:

cos (0) = 1

cos (0)

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}

= 1

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

بسّط

= \frac{1}{2}

\frac{1 + cos ( 4 \cdot 0 )}{8} = \frac{1}{4} (Decimal : 0.25)

\frac{1 + cos ( 4 \cdot 0 )}{8}

بسّط

= \frac{1 + cos ( 0 )}{8}

استخدم المتطابقة الأساسيّة التالية:

cos (0) = 1

cos (0)

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}

= 1

= \frac{1 + 1}{8}

\frac{1 + 1}{8}

بسّط:

\frac{1}{4}

\frac{1 + 1}{8}

1 + 1 = 2 :

اجمع الأعداد

= \frac{2}{8}

2 :

إلغ العوامل المشتركة

= \frac{1}{4}

= \frac{1}{4}

لا يتساوى الطرفان

\Rightarrow خطأ

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