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

الحلّ

2 cos (x) = sin (2 x)

الحلّ

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

+1

درجات

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

خطوات الحلّ

2 cos (x) = sin (2 x)

من الطرفين

sin (2 x)

اطرح

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

Rewrite using trig identities

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

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

:فعّل متطابقة الزاوية المضاعفة

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

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

cos (x) 2 - 2 cos (x) sin (x)

حلل إلى عوامل:

cos (x) (2 - 2 sin (x))

cos (x) 2 - 2 cos (x) sin (x)

cos (x)

قم باخراج العامل المشترك

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

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

حلّ كل جزء على حدة

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

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

cos (x) = 0

cos (x) = 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}

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

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

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

2 - 2 sin (x) = 0

انقل

2

إلى الجانب الأيمن

2 - 2 sin (x) = 0

من الطرفين

2

اطرح

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

بسّط

- 2 sin (x) = - 2

- 2 sin (x) = - 2

- 2

اقسم الطرفين على

- 2 sin (x) = - 2

- 2

اقسم الطرفين على

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

بسّط

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

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

sin (x) = \frac{2}{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}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

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

وحّد الحلول

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

أمثلة شائعة

cos (θ) = 12

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

cos^{2} (x) + 2 cos (x) = 0

2sin(x)=tan(2x)

2 sin (x) = tan (2 x)

3sin(x)-4cos(x)=-2

3 sin (x) - 4 cos (x) = - 2

6sin(x)+cos^2(x)=2

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