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شائع >

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

الحلّ

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

الحلّ

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

+1

درجات

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

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

من الطرفين

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

اطرح

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

Rewrite using trig identities

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

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

:فعّل متطابقة الزاوية المضاعفة

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

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

بسّط:

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

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

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

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

2^{2} = 4

= 4 sin^{2} (x) cos^{2} (x) - 2 cos^{2} (x) sin^{2} (x)

4 sin^{2} (x) cos^{2} (x) - 2 cos^{2} (x) sin^{2} (x) = 2 sin^{2} (x) cos^{2} (x) :

اجمع العناصر المتشابهة

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

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

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

حلّ كل جزء على حدة

cos^{2} (x) = 0 or sin^{2} (x) = 0

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

cos^{2} (x) = 0

x^{n} = 0 \Rightarrow x = 0

فعّل القانون

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

sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

sin^{2} (x) = 0

x^{n} = 0 \Rightarrow x = 0

فعّل القانون

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

وحّد الحلول

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

رسم

أمثلة شائعة

tan (θ) = - \frac{1}{5}

(sin(70))/(25)=(sin(c))/(26)

\frac{sin ( 7 0 ^{\circ} )}{25} = \frac{sin ( c )}{26}

\frac{csc ( x )}{2} = 3

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

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

1.25sin(30)=1.5sin(x)

1.25 sin (3 0^{\circ}) = 1.5 sin (x)