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

الحلّ

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

الحلّ

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

+1

درجات

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = - 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 194.47751 \dots^{\circ} + 36 0^{\circ} n

خطوات الحلّ

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

Rewrite using trig identities

cos (x) + 2 sin (2 x)

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

:فعّل متطابقة الزاوية المضاعفة

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

بسّط

= cos (x) + 4 sin (x) cos (x)

cos (x) + 4 cos (x) sin (x) = 0

cos (x) + 4 cos (x) sin (x)

حلل إلى عوامل:

cos (x) (4 sin (x) + 1)

cos (x) + 4 cos (x) sin (x)

cos (x)

قم باخراج العامل المشترك

= cos (x) (1 + 4 sin (x))

cos (x) (4 sin (x) + 1) = 0

حلّ كل جزء على حدة

cos (x) = 0 or 4 sin (x) + 1 = 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

4 sin (x) + 1 = 0 : x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

4 sin (x) + 1 = 0

انقل

1

إلى الجانب الأيمن

4 sin (x) + 1 = 0

من الطرفين

1

اطرح

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

بسّط

4 sin (x) = - 1

4 sin (x) = - 1

4

اقسم الطرفين على

4 sin (x) = - 1

4

اقسم الطرفين على

\frac{4 sin ( x )}{4} = \frac{- 1}{4}

بسّط

sin (x) = - \frac{1}{4}

sin (x) = - \frac{1}{4}

Apply trig inverse properties

sin (x) = - \frac{1}{4}

sin (x) = - \frac{1}{4} :

حلول عامّة لـ

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

وحّد الحلول

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

أظهر الحلّ بالتمثيل العشريّ

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

أمثلة شائعة

sin (d) = 0.81

sec (\frac{5 x}{4}) = 2

solvefor t,16sin(t)cos(t)=6sin(t)solve for

t, 16 sin (t) cos (t) = 6 sin (t)

6cos(6t)+24sin(6t)=0

6 cos (6 t) + 24 sin (6 t) = 0

sin(x)=(50*sin(145))/(120)

sin (x) = \frac{50 \cdot sin ( 14 5 ^{\circ} )}{120}