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شائع علم المثلثات >

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

الحلّ

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

الحلّ

x = π + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

درجات

x = 18 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

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

من الطرفين

sin^{2} (x)

اطرح

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

Rewrite using trig identities

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

cos^{2} (x) + sin^{2} (x) = 1

:فعّل نطريّة فيتاغوروس

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

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

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

بسّط:

- cos (x) + 2 sin (x) + 2 sin (x) cos (x) - 1

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

- (1 - cos^{2} (x)) : - 1 + cos^{2} (x)

- (1 - cos^{2} (x))

افتح أقواس

= - (1) - (- cos^{2} (x))

فعّل قوانين سالب-موجب

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

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

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

(- cos (x) + 2 sin (x)) (1 + cos (x))

وسٌع:

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

(- cos (x) + 2 sin (x)) (1 + cos (x))

(a + b) (c + d) = a c + a d + b c + b d

(a + b) (c + d) = a c + a d + b c + b d

فعّل قانون التوزيع

a = - cos (x), b = 2 sin (x), c = 1, d = cos (x)

= (- cos (x)) \cdot 1 + (- cos (x)) cos (x) + 2 sin (x) \cdot 1 + 2 sin (x) cos (x)

فعّل قوانين سالب-موجب

+ (- a) = - a

= - 1 \cdot cos (x) - cos (x) cos (x) + 2 \cdot 1 \cdot sin (x) + 2 sin (x) cos (x)

- 1 \cdot cos (x) - cos (x) cos (x) + 2 \cdot 1 \cdot sin (x) + 2 sin (x) cos (x)

بسّط:

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

- 1 \cdot cos (x) - cos (x) cos (x) + 2 \cdot 1 \cdot sin (x) + 2 sin (x) cos (x)

1 \cdot cos (x) = cos (x)

1 \cdot cos (x)

1 \cdot cos (x) = cos (x) :

اضرب

= cos (x)

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

cos (x) cos (x)

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

cos (x) cos (x) = cos^{1 + 1} (x)

= cos^{1 + 1} (x)

1 + 1 = 2 :

اجمع الأعداد

= cos^{2} (x)

2 \cdot 1 \cdot sin (x) = 2 sin (x)

2 \cdot 1 \cdot sin (x)

2 \cdot 1 = 2 :

اضرب الأعداد

= 2 sin (x)

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

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

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

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

بسّط:

- cos (x) + 2 sin (x) + 2 sin (x) cos (x) - 1

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

جمّع التعابير المتشابهة

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

cos^{2} (x) - cos^{2} (x) = 0 :

اجمع العناصر المتشابهة

= - cos (x) + 2 sin (x) + 2 sin (x) cos (x) - 1

= - cos (x) + 2 sin (x) + 2 sin (x) cos (x) - 1

= - cos (x) + 2 sin (x) + 2 sin (x) cos (x) - 1

- 1 - cos (x) + 2 sin (x) + 2 cos (x) sin (x) = 0

- 1 - cos (x) + 2 sin (x) + 2 cos (x) sin (x)

حلل إلى عوامل:

(cos (x) + 1) (2 sin (x) - 1)

- 1 - cos (x) + 2 sin (x) + 2 cos (x) sin (x)

= (- cos (x) - 1) + (2 sin (x) cos (x) + 2 sin (x))

2 sin (x) (cos (x) + 1)

2 sin (x) cos (x) + 2 sin (x)

من

2 sin (x)

اخرج العامل

2 sin (x) cos (x) + 2 sin (x)

2 sin (x)

قم باخراج العامل المشترك

= 2 sin (x) (cos (x) + 1)

- (cos (x) + 1)

- cos (x) - 1

من

- 1

اخرج العامل

- cos (x) - 1

- 1

قم باخراج العامل المشترك

= - (cos (x) + 1)

= 2 sin (x) (cos (x) + 1) - (cos (x) + 1)

cos (x) + 1

قم باخراج العامل المشترك

= (cos (x) + 1) (2 sin (x) - 1)

(cos (x) + 1) (2 sin (x) - 1) = 0

حلّ كل جزء على حدة

cos (x) + 1 = 0 or 2 sin (x) - 1 = 0

cos (x) + 1 = 0 : x = π + 2 πn

cos (x) + 1 = 0

انقل

1

إلى الجانب الأيمن

cos (x) + 1 = 0

من الطرفين

1

اطرح

cos (x) + 1 - 1 = 0 - 1

بسّط

cos (x) = - 1

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

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 sin (x) - 1 = 0

انقل

1

إلى الجانب الأيمن

2 sin (x) - 1 = 0

للطرفين

1

أضف

2 sin (x) - 1 + 1 = 0 + 1

بسّط

2 sin (x) = 1

2 sin (x) = 1

2

اقسم الطرفين على

2 sin (x) = 1

2

اقسم الطرفين على

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

بسّط

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

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

sin (x) = \frac{1}{2} :

حلول عامّة لـ

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}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

وحّد الحلول

x = π + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

رسم

أمثلة شائعة

sec^2(x)+2tan^2(x)=2

sec^{2} (x) + 2 tan^{2} (x) = 2

sin(x^2-2x)=0

sin (x^{2} - 2 x) = 0

8sin^2(x)+4cos^2(x)=7

8 sin^{2} (x) + 4 cos^{2} (x) = 7

10cos^2(x)+cos(x)=11sin^2(x)-9

10 cos^{2} (x) + cos (x) = 11 sin^{2} (x) - 9

sin(6.5x+2.5)=0.0851

sin (6.5 x + 2.5) = 0.0851