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

الحلّ

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

الحلّ

x = \frac{π}{12} + \frac{2 πn}{3}, x = \frac{7 π}{12} + \frac{2 πn}{3}

درجات

x = 1 5^{\circ} + 12 0^{\circ} n, x = 10 5^{\circ} + 12 0^{\circ} n

خطوات الحلّ

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

Rewrite using trig identities

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

cos (s) cos (t) - sin (s) sin (t) = cos (s + t)

:فعّل متطابقة الجمع لزوايا

= cos (2 x + x)

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

cos (2 x + 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}

2 x + x = \frac{π}{4} + 2 πn, 2 x + x = \frac{7 π}{4} + 2 πn

2 x + x = \frac{π}{4} + 2 πn, 2 x + x = \frac{7 π}{4} + 2 πn

2 x + x = \frac{π}{4} + 2 πn

حلّ:

x = \frac{π}{12} + \frac{2 πn}{3}

2 x + x = \frac{π}{4} + 2 πn

2 x + x = 3 x :

اجمع العناصر المتشابهة

3 x = \frac{π}{4} + 2 πn

3

اقسم الطرفين على

3 x = \frac{π}{4} + 2 πn

3

اقسم الطرفين على

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

بسّط

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

\frac{3 x}{3}

بسّط:

x

\frac{3 x}{3}

\frac{3}{3} = 1 :

اقسم الأعداد

= x

\frac{\frac{π}{4}}{3} + \frac{2 πn}{3}

بسّط:

\frac{π}{12} + \frac{2 πn}{3}

\frac{\frac{π}{4}}{3} + \frac{2 πn}{3}

\frac{\frac{π}{4}}{3} = \frac{π}{12}

\frac{\frac{π}{4}}{3}

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

: استخدم ميزات الكسور التالية

= \frac{π}{4 \cdot 3}

4 \cdot 3 = 12 :

اضرب الأعداد

= \frac{π}{12}

= \frac{π}{12} + \frac{2 πn}{3}

x = \frac{π}{12} + \frac{2 πn}{3}

x = \frac{π}{12} + \frac{2 πn}{3}

x = \frac{π}{12} + \frac{2 πn}{3}

2 x + x = \frac{7 π}{4} + 2 πn

حلّ:

x = \frac{7 π}{12} + \frac{2 πn}{3}

2 x + x = \frac{7 π}{4} + 2 πn

2 x + x = 3 x :

اجمع العناصر المتشابهة

3 x = \frac{7 π}{4} + 2 πn

3

اقسم الطرفين على

3 x = \frac{7 π}{4} + 2 πn

3

اقسم الطرفين على

\frac{3 x}{3} = \frac{\frac{7 π}{4}}{3} + \frac{2 πn}{3}

بسّط

\frac{3 x}{3} = \frac{\frac{7 π}{4}}{3} + \frac{2 πn}{3}

\frac{3 x}{3}

بسّط:

x

\frac{3 x}{3}

\frac{3}{3} = 1 :

اقسم الأعداد

= x

\frac{\frac{7 π}{4}}{3} + \frac{2 πn}{3}

بسّط:

\frac{7 π}{12} + \frac{2 πn}{3}

\frac{\frac{7 π}{4}}{3} + \frac{2 πn}{3}

\frac{\frac{7 π}{4}}{3} = \frac{7 π}{12}

\frac{\frac{7 π}{4}}{3}

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

: استخدم ميزات الكسور التالية

= \frac{7 π}{4 \cdot 3}

4 \cdot 3 = 12 :

اضرب الأعداد

= \frac{7 π}{12}

= \frac{7 π}{12} + \frac{2 πn}{3}

x = \frac{7 π}{12} + \frac{2 πn}{3}

x = \frac{7 π}{12} + \frac{2 πn}{3}

x = \frac{7 π}{12} + \frac{2 πn}{3}

x = \frac{π}{12} + \frac{2 πn}{3}, x = \frac{7 π}{12} + \frac{2 πn}{3}

أمثلة شائعة

-2+cos(θ)=-2+6/5*cos(θ)

- 2 + cos (θ) = - 2 + \frac{6}{5} \cdot cos (θ)

tan^2(y)+3-2sec^2(y)=0

tan^{2} (y) + 3 - 2 sec^{2} (y) = 0

500=538.5cos(x-10)

500 = 538.5 cos (x - 1 0^{\circ})

-2cos^2(x)=3(sin(x)-1)

- 2 cos^{2} (x) = 3 (sin (x) - 1)

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

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