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3/(cos^2(x))=3+2tan^2(x)

الحلّ

\frac{3}{cos ^{2} ( x )} = 3 + 2 tan^{2} (x)

الحلّ

x = 2 πn, x = π + 2 πn

درجات

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

خطوات الحلّ

\frac{3}{cos ^{2} ( x )} = 3 + 2 tan^{2} (x)

من الطرفين

3 + 2 tan^{2} (x)

اطرح

\frac{3}{cos ^{2} ( x )} - 3 - 2 tan^{2} (x) = 0

\frac{3}{cos ^{2} ( x )} - 3 - 2 tan^{2} (x)

بسّط:

\frac{3 - 3 cos ^{2} ( x ) - 2 tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

\frac{3}{cos ^{2} ( x )} - 3 - 2 tan^{2} (x)

3 = \frac{3 cos ^{2} ( x )}{cos ^{2} ( x )}, 2 tan^{2} (x) = \frac{2 tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

:حوّل الأعداد لكسور

= \frac{3}{cos ^{2} ( x )} - \frac{3 cos ^{2} ( x )}{cos ^{2} ( x )} - \frac{2 tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{3 - 3 cos ^{2} ( x ) - 2 tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

\frac{3 - 3 cos ^{2} ( x ) - 2 tan ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 - 3 cos^{2} (x) - 2 tan^{2} (x) cos^{2} (x) = 0

sin, cos :

عبّر بواسطة

3 - 3 cos^{2} (x) - 2 (\frac{sin ( x )}{cos ( x )})^{2} cos^{2} (x) = 0

3 - 3 cos^{2} (x) - 2 (\frac{sin ( x )}{cos ( x )})^{2} cos^{2} (x)

بسّط:

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

3 - 3 cos^{2} (x) - 2 (\frac{sin ( x )}{cos ( x )})^{2} cos^{2} (x)

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

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

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

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

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

:فعّل قانون القوى

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

= 2 \cdot \frac{sin ^{2} ( x )}{cos ^{2} ( x )} cos^{2} (x)

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

:اضرب كسور

= \frac{sin ^{2} ( x ) \cdot 2 cos ^{2} ( x )}{cos ^{2} ( x )}

cos^{2} (x) :

إلغ العوامل المشتركة

= sin^{2} (x) \cdot 2

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

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

للطرفين

2 sin^{2} (x)

أضف

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

ربّع الطرفين

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

من الطرفين

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

اطرح

(3 - 3 cos^{2} (x))^{2} - 4 sin^{4} (x) = 0

(3 - 3 cos^{2} (x))^{2} - 4 sin^{4} (x)

حلل إلى عوامل:

(3 - 3 cos^{2} (x) + 2 sin^{2} (x)) (3 - 3 cos^{2} (x) - 2 sin^{2} (x))

(3 - 3 cos^{2} (x))^{2} - 4 sin^{4} (x)

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

كـ

(3 - 3 cos^{2} (x))^{2} - 4 sin^{4} (x)

اكتب مجددًا

(3 - 3 cos^{2} (x))^{2} - 4 sin^{4} (x)

2^{2}

كـ

4

اكتب مجددًا

= (3 - 3 cos^{2} (x))^{2} - 2^{2} sin^{4} (x)

a^{b c} = (a^{b})^{c}

:فعّل قانون القوى

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

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

a^{m} b^{m} = (ab)^{m}

:فعّل قانون القوى

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

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

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

x^{2} - y^{2} = (x + y) (x - y)

فعّل قانون فرق المربّعات

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

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

بسّط

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

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

حلّ كل جزء على حدة

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

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

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

Rewrite using trig identities

3 + 2 sin^{2} (x) - 3 cos^{2} (x)

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

:فعّل نطريّة فيتاغوروس

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

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

3 + 2 sin^{2} (x) - 3 (1 - sin^{2} (x))

بسّط:

5 sin^{2} (x)

3 + 2 sin^{2} (x) - 3 (1 - sin^{2} (x))

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

وسٌع:

- 3 + 3 sin^{2} (x)

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

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = - 3, b = 1, c = sin^{2} (x)

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

فعّل قوانين سالب-موجب

- (- a) = a

= - 3 \cdot 1 + 3 sin^{2} (x)

3 \cdot 1 = 3 :

اضرب الأعداد

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

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

3 + 2 sin^{2} (x) - 3 + 3 sin^{2} (x)

بسّط:

5 sin^{2} (x)

3 + 2 sin^{2} (x) - 3 + 3 sin^{2} (x)

جمّع التعابير المتشابهة

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

2 sin^{2} (x) + 3 sin^{2} (x) = 5 sin^{2} (x) :

اجمع العناصر المتشابهة

= 5 sin^{2} (x) + 3 - 3

3 - 3 = 0

= 5 sin^{2} (x)

= 5 sin^{2} (x)

= 5 sin^{2} (x)

5 sin^{2} (x) = 0

5

اقسم الطرفين على

5 sin^{2} (x) = 0

5

اقسم الطرفين على

5 sin^{2} (x) = 0

5

اقسم الطرفين على

\frac{5 sin ^{2} ( x )}{5} = \frac{0}{5}

بسّط

sin^{2} (x) = 0

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

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

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

بسّط:

sin^{2} (x)

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

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

وسٌع:

- 3 + 3 sin^{2} (x)

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

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = - 3, b = 1, c = sin^{2} (x)

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

فعّل قوانين سالب-موجب

- (- a) = a

= - 3 \cdot 1 + 3 sin^{2} (x)

3 \cdot 1 = 3 :

اضرب الأعداد

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

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

3 - 2 sin^{2} (x) - 3 + 3 sin^{2} (x)

بسّط:

sin^{2} (x)

3 - 2 sin^{2} (x) - 3 + 3 sin^{2} (x)

جمّع التعابير المتشابهة

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

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

اجمع العناصر المتشابهة

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

3 - 3 = 0

= sin^{2} (x)

= sin^{2} (x)

= sin^{2} (x)

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 = 2 πn, x = π + 2 πn

تأكّد من صحّة الحلول عن طريق تعويضها في المعادلة الأصليّة

للتحقّق من دقّة الحلول

\frac{3}{cos ^{2} ( x )} = 3 + 2 tan^{2} (x)

عوّض الحلول في

إلغي الحلول التي تعطي قضيّة كذب

2 πn

افحص الحل:

صحيح

2 πn

n = 1

استبدل

2 π 1

x = 2 π 1

عوّض

, \frac{3}{cos ^{2} ( x )} = 3 + 2 tan^{2} (x)

في

\frac{3}{cos ^{2} ( 2 π 1 )} = 3 + 2 tan^{2} (2 π 1)

بسّط

3 = 3

\Rightarrow صحيح

π + 2 πn

افحص الحل:

صحيح

π + 2 πn

n = 1

استبدل

π + 2 π 1

x = π + 2 π 1

عوّض

, \frac{3}{cos ^{2} ( x )} = 3 + 2 tan^{2} (x)

في

\frac{3}{cos ^{2} ( π + 2 π 1 )} = 3 + 2 tan^{2} (π + 2 π 1)

بسّط

3 = 3

\Rightarrow صحيح

x = 2 πn, x = π + 2 πn

أمثلة شائعة

13*sin(x)=12

13 \cdot sin (x) = 12

11sin(θ)=11cos(θ)

11 sin (θ) = 11 cos (θ)

1-2tan(x)=0

1 - 2 tan (x) = 0

sin(a)=0.747

sin (a) = 0.747

1-tan^2(y)=2sec(y)-4

1 - tan^{2} (y) = 2 sec (y) - 4