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شائع علم المثلثات >

cos(x)=1-sin^2(x/2)

الحلّ

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

الحلّ

x = 4 πn, x = 2 π + 4 πn

+1

درجات

x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

خطوات الحلّ

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

من الطرفين

1 - sin^{2} (\frac{x}{2})

اطرح

cos (x) - 1 + sin^{2} (\frac{x}{2}) = 0

Rewrite using trig identities

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

cos^{2} (x) - sin^{2} (x) = cos (2 x)

:فعّل متطابقة الزاوية المضاعفة

- sin^{2} (x) = cos (2 x) - cos^{2} (x)

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

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

بسّط:

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

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

2 \cdot \frac{x}{2}

اضرب بـ:

x

2 \cdot \frac{x}{2}

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

:اضرب كسور

= \frac{x \cdot 2}{2}

2 :

إلغ العوامل المشتركة

= x

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

- (cos (x) - cos^{2} (\frac{x}{2})) : - cos (x) + cos^{2} (\frac{x}{2})

- (cos (x) - cos^{2} (\frac{x}{2}))

افتح أقواس

= - (cos (x)) - (- cos^{2} (\frac{x}{2}))

فعّل قوانين سالب-موجب

- (- a) = a, - (a) = - a

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

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

cos (x) - cos (x) = 0 :

اجمع العناصر المتشابهة

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

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

- 1 + cos^{2} (\frac{x}{2}) = 0

بالاستعانة بطريقة التعويض

- 1 + cos^{2} (\frac{x}{2}) = 0

cos (\frac{x}{2}) = u :

على افتراض أنّ

- 1 + u^{2} = 0

- 1 + u^{2} = 0 : u = 1, u = - 1

- 1 + u^{2} = 0

انقل

1

إلى الجانب الأيمن

- 1 + u^{2} = 0

للطرفين

1

أضف

- 1 + u^{2} + 1 = 0 + 1

بسّط

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 (\frac{x}{2})

استبدل مجددًا

cos (\frac{x}{2}) = 1, cos (\frac{x}{2}) = - 1

cos (\frac{x}{2}) = 1, cos (\frac{x}{2}) = - 1

cos (\frac{x}{2}) = 1 : x = 4 πn

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

cos (\frac{x}{2}) = 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}

\frac{x}{2} = 0 + 2 πn

\frac{x}{2} = 0 + 2 πn

\frac{x}{2} = 0 + 2 πn

حلّ:

x = 4 πn

\frac{x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{2} = 2 πn

2

اضرب الطرفين بـ

\frac{x}{2} = 2 πn

2

اضرب الطرفين بـ

\frac{2 x}{2} = 2 \cdot 2 πn

بسّط

x = 4 πn

x = 4 πn

x = 4 πn

cos (\frac{x}{2}) = - 1 : x = 2 π + 4 πn

cos (\frac{x}{2}) = - 1

cos (\frac{x}{2}) = - 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}

\frac{x}{2} = π + 2 πn

\frac{x}{2} = π + 2 πn

\frac{x}{2} = π + 2 πn

حلّ:

x = 2 π + 4 πn

\frac{x}{2} = π + 2 πn

2

اضرب الطرفين بـ

\frac{x}{2} = π + 2 πn

2

اضرب الطرفين بـ

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

بسّط

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 2 π + 4 πn

وحّد الحلول

x = 4 πn, x = 2 π + 4 πn

رسم

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