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شائع علم المثلثات >

cos^2(x)= 1/2+1/2 cos^2(x)

الحلّ

cos^{2} (x) = \frac{1}{2} + \frac{1}{2} cos^{2} (x)

الحلّ

x = 2 πn, x = π + 2 πn

+1

درجات

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

cos^{2} (x) = \frac{1}{2} + \frac{1}{2} cos^{2} (x)

بالاستعانة بطريقة التعويض

cos^{2} (x) = \frac{1}{2} + \frac{1}{2} cos^{2} (x)

cos (x) = u :

على افتراض أنّ

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

u^{2} = \frac{1}{2} + \frac{1}{2} u^{2} : u = 1, u = - 1

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

انقل

\frac{1}{2} u^{2}

إلى الجانب الأيسر

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

من الطرفين

\frac{1}{2} u^{2}

اطرح

u^{2} - \frac{1}{2} u^{2} = \frac{1}{2} + \frac{1}{2} u^{2} - \frac{1}{2} u^{2}

بسّط

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

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

2

اضرب الطرفين بـ

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

2

اضرب الطرفين بـ

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

بسّط

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

2 \cdot \frac{1}{2} u^{2}

بسّط:

u^{2}

2 \cdot \frac{1}{2} u^{2}

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= u^{2} \cdot 1

u^{2} \cdot 1 = u^{2} :

اضرب

= u^{2}

\frac{1 \cdot 2}{2}

بسّط:

1

\frac{1 \cdot 2}{2}

1 \cdot 2 = 2 :

اضرب الأعداد

= \frac{2}{2}

\frac{a}{a} = 1

فعّل القانون

= 1

u^{2} = 1

u^{2} = 1

u^{2} = 1

x = f (a), - f (a)

الحلول هي

x^{2} = f (a)

لـ

u = 1, u = - 1

1 = 1

1

1 = 1

فعّل القانون

= 1

- 1 = - 1

- 1

1 = 1

فعّل القانون

= - 1

u = 1, u = - 1

u = cos (x)

استبدل مجددًا

cos (x) = 1, cos (x) = - 1

cos (x) = 1, cos (x) = - 1

cos (x) = 1 : x = 2 πn

cos (x) = 1

cos (x) = 1 :

حلول عامّة لـ

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}

x = 0 + 2 πn

x = 0 + 2 πn

x = 0 + 2 πn

حلّ:

x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

cos (x) = - 1 :

حلول عامّة لـ

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}

x = π + 2 πn

x = π + 2 πn

وحّد الحلول

x = 2 πn, x = π + 2 πn

رسم

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