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cos^2(2x)=1

الحلّ

cos^{2} (2 x) = 1

الحلّ

x = πn, x = \frac{π}{2} + πn

+1

درجات

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

خطوات الحلّ

cos^{2} (2 x) = 1

بالاستعانة بطريقة التعويض

cos^{2} (2 x) = 1

cos (2 x) = u :

على افتراض أنّ

u^{2} = 1

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

x = f (a), - f (a)

الحلول هي

x^{2} = f (a)

لـ

u = 1, u = - 1

1 = 1

1

1 = 1

فعّل القانون

= 1

- 1 = - 1

- 1

1 = 1

فعّل القانون

= - 1

u = 1, u = - 1

u = cos (2 x)

استبدل مجددًا

cos (2 x) = 1, cos (2 x) = - 1

cos (2 x) = 1, cos (2 x) = - 1

cos (2 x) = 1 : x = πn

cos (2 x) = 1

cos (2 x) = 1 :

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 = 0 + 2 πn

2 x = 0 + 2 πn

2 x = 0 + 2 πn

حلّ:

x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

2

اقسم الطرفين على

2 x = 2 πn

2

اقسم الطرفين على

\frac{2 x}{2} = \frac{2 πn}{2}

بسّط

x = πn

x = πn

x = πn

cos (2 x) = - 1 : x = \frac{π}{2} + πn

cos (2 x) = - 1

cos (2 x) = - 1 :

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 = π + 2 πn

2 x = π + 2 πn

2 x = π + 2 πn

حلّ:

x = \frac{π}{2} + πn

2 x = π + 2 πn

2

اقسم الطرفين على

2 x = π + 2 πn

2

اقسم الطرفين على

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

بسّط

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

وحّد الحلول

x = πn, x = \frac{π}{2} + πn

أمثلة شائعة

- 4 cos^{2} (θ) = - 1

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

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

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

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

cos(x)= 1/(sqrt(10))

cos (x) = \frac{1}{10}

solvefor u,x=5sinh(u)solve for

u, x = 5 sinh (u)