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شائع علم المثلثات >

2sin^2(x)-4cos^2(x)+1=0

الحلّ

2 sin^{2} (x) - 4 cos^{2} (x) + 1 = 0

الحلّ

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

درجات

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

خطوات الحلّ

2 sin^{2} (x) - 4 cos^{2} (x) + 1 = 0

Rewrite using trig identities

1 + 2 sin^{2} (x) - 4 cos^{2} (x)

cos^{2} (x) + sin^{2} (x) = 1

:فعّل نطريّة فيتاغوروس

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

= 1 + 2 sin^{2} (x) - 4 (1 - sin^{2} (x))

1 + 2 sin^{2} (x) - 4 (1 - sin^{2} (x))

بسّط:

6 sin^{2} (x) - 3

1 + 2 sin^{2} (x) - 4 (1 - sin^{2} (x))

- 4 (1 - sin^{2} (x))

وسٌع:

- 4 + 4 sin^{2} (x)

- 4 (1 - sin^{2} (x))

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = - 4, b = 1, c = sin^{2} (x)

= - 4 \cdot 1 - (- 4) sin^{2} (x)

فعّل قوانين سالب-موجب

- (- a) = a

= - 4 \cdot 1 + 4 sin^{2} (x)

4 \cdot 1 = 4 :

اضرب الأعداد

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

= 1 + 2 sin^{2} (x) - 4 + 4 sin^{2} (x)

1 + 2 sin^{2} (x) - 4 + 4 sin^{2} (x)

بسّط:

6 sin^{2} (x) - 3

1 + 2 sin^{2} (x) - 4 + 4 sin^{2} (x)

جمّع التعابير المتشابهة

= 2 sin^{2} (x) + 4 sin^{2} (x) + 1 - 4

2 sin^{2} (x) + 4 sin^{2} (x) = 6 sin^{2} (x) :

اجمع العناصر المتشابهة

= 6 sin^{2} (x) + 1 - 4

1 - 4 = - 3 :

اطرح/اجمع الأعداد

= 6 sin^{2} (x) - 3

= 6 sin^{2} (x) - 3

= 6 sin^{2} (x) - 3

- 3 + 6 sin^{2} (x) = 0

بالاستعانة بطريقة التعويض

- 3 + 6 sin^{2} (x) = 0

sin (x) = u :

على افتراض أنّ

- 3 + 6 u^{2} = 0

- 3 + 6 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

- 3 + 6 u^{2} = 0

انقل

3

إلى الجانب الأيمن

- 3 + 6 u^{2} = 0

للطرفين

3

أضف

- 3 + 6 u^{2} + 3 = 0 + 3

بسّط

6 u^{2} = 3

6 u^{2} = 3

6

اقسم الطرفين على

6 u^{2} = 3

6

اقسم الطرفين على

\frac{6 u ^{2}}{6} = \frac{3}{6}

بسّط

u^{2} = \frac{1}{2}

u^{2} = \frac{1}{2}

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

الحلول هي

x^{2} = f (a)

لـ

u = \frac{1}{2}, u = - \frac{1}{2}

u = sin (x)

استبدل مجددًا

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

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

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{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

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sin (x) = - \frac{1}{2}

sin (x) = - \frac{1}{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{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

وحّد الحلول

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

رسم

أمثلة شائعة

solvefor x,-y^2-cos(x)=0

so l v e f or x, - y^{2} - cos (x) = 0

cos(θ)= 6/16

cos (θ) = \frac{6}{16}

|cos(x)|=-1

∣ cos (x) ∣ = - 1

(sin(25)}{5.41}=\frac{sin(x))/9

\frac{sin ( 2 5 ^{\circ} )}{5.41} = \frac{sin ( x )}{9}

cos(θ)= 4/(sqrt(17))

cos (θ) = \frac{4}{17}