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شائع علم المثلثات >

4cos^2(t)sin(t)-2cos(t)sin(t)=0

الحلّ

4 cos^{2} (t) sin (t) - 2 cos (t) sin (t) = 0

الحلّ

t = 2 πn, t = π + 2 πn, t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn, t = \frac{π}{3} + 2 πn, t = \frac{5 π}{3} + 2 πn

+1

درجات

t = 0^{\circ} + 36 0^{\circ} n, t = 18 0^{\circ} + 36 0^{\circ} n, t = 9 0^{\circ} + 36 0^{\circ} n, t = 27 0^{\circ} + 36 0^{\circ} n, t = 6 0^{\circ} + 36 0^{\circ} n, t = 30 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

4 cos^{2} (t) sin (t) - 2 cos (t) sin (t) = 0

4 cos^{2} (t) sin (t) - 2 cos (t) sin (t)

حلل إلى عوامل:

2 sin (t) cos (t) (2 cos (t) - 1)

4 cos^{2} (t) sin (t) - 2 cos (t) sin (t)

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

:فعّل قانون القوى

sin (t) cos^{2} (t) = sin (t) cos (t) cos (t)

= 4 sin (t) cos (t) cos (t) - 2 sin (t) cos (t)

2 \cdot 2

كـ

4

اكتب مجددًا

= 2 \cdot 2 sin (t) cos (t) cos (t) - 2 sin (t) cos (t)

2 sin (t) cos (t)

قم باخراج العامل المشترك

= 2 sin (t) cos (t) (2 cos (t) - 1)

2 sin (t) cos (t) (2 cos (t) - 1) = 0

حلّ كل جزء على حدة

sin (t) = 0 or cos (t) = 0 or 2 cos (t) - 1 = 0

sin (t) = 0 : t = 2 πn, t = π + 2 πn

sin (t) = 0

sin (t) = 0 :

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}

t = 0 + 2 πn, t = π + 2 πn

t = 0 + 2 πn, t = π + 2 πn

t = 0 + 2 πn

حلّ:

t = 2 πn

t = 0 + 2 πn

0 + 2 πn = 2 πn

t = 2 πn

t = 2 πn, t = π + 2 πn

cos (t) = 0 : t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn

cos (t) = 0

cos (t) = 0 :

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}

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn

2 cos (t) - 1 = 0 : t = \frac{π}{3} + 2 πn, t = \frac{5 π}{3} + 2 πn

2 cos (t) - 1 = 0

انقل

1

إلى الجانب الأيمن

2 cos (t) - 1 = 0

للطرفين

1

أضف

2 cos (t) - 1 + 1 = 0 + 1

بسّط

2 cos (t) = 1

2 cos (t) = 1

2

اقسم الطرفين على

2 cos (t) = 1

2

اقسم الطرفين على

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

بسّط

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

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

cos (t) = \frac{1}{2} :

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}

t = \frac{π}{3} + 2 πn, t = \frac{5 π}{3} + 2 πn

t = \frac{π}{3} + 2 πn, t = \frac{5 π}{3} + 2 πn

وحّد الحلول

t = 2 πn, t = π + 2 πn, t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn, t = \frac{π}{3} + 2 πn, t = \frac{5 π}{3} + 2 πn

رسم

أمثلة شائعة

3(1-sin(θ))=2cos^2(θ)

3 (1 - sin (θ)) = 2 cos^{2} (θ)

sin(x)=(sqrt(5))/5

sin (x) = \frac{5}{5}

tan(θ)=0.11246

tan (θ) = 0.11246

cos(2θ)-sin^2(θ)=1

cos (2 θ) - sin^{2} (θ) = 1

sin(x)=sqrt(3/4)

sin (x) = \frac{3}{4}