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sin(2x)=-sqrt(2)sin(x)

الحلّ

sin (2 x) = - 2 sin (x)

الحلّ

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

+1

درجات

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n

خطوات الحلّ

sin (2 x) = - 2 sin (x)

من الطرفين

- 2 sin (x)

اطرح

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

Rewrite using trig identities

sin (2 x) + sin (x) 2

sin (2 x) = 2 sin (x) cos (x)

:فعّل متطابقة الزاوية المضاعفة

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

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

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

حلل إلى عوامل:

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

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

sin (x)

قم باخراج العامل المشترك

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

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

حلّ كل جزء على حدة

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

2 + 2 cos (x) = 0 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

2 + 2 cos (x) = 0

انقل

2

إلى الجانب الأيمن

2 + 2 cos (x) = 0

من الطرفين

2

اطرح

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

بسّط

2 cos (x) = - 2

2 cos (x) = - 2

2

اقسم الطرفين على

2 cos (x) = - 2

2

اقسم الطرفين على

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

بسّط

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

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

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

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

وحّد الحلول

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

أمثلة شائعة

10sec^2(x)+5tan^2(x)-15=0

10 sec^{2} (x) + 5 tan^{2} (x) - 15 = 0

sin (2 x) + 1 = 0

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

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

cos(2θ)+10sin^2(θ)=7

cos (2 θ) + 10 sin^{2} (θ) = 7

4 cos (2 t) + 1 = 3