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شائع >

3sec^2(x)tan(x)=4tan(x)

الحلّ

3 sec^{2} (x) tan (x) = 4 tan (x)

الحلّ

x = πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

درجات

x = 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

3 sec^{2} (x) tan (x) = 4 tan (x)

من الطرفين

4 tan (x)

اطرح

3 sec^{2} (x) tan (x) - 4 tan (x) = 0

3 sec^{2} (x) tan (x) - 4 tan (x)

حلل إلى عوامل:

tan (x) (3 sec (x) + 2) (3 sec (x) - 2)

3 sec^{2} (x) tan (x) - 4 tan (x)

tan (x)

قم باخراج العامل المشترك

= tan (x) (3 sec^{2} (x) - 4)

3 sec^{2} (x) - 4

حلل إلى عوامل:

(3 sec (x) + 2) (3 sec (x) - 2)

3 sec^{2} (x) - 4

(3 sec (x))^{2} - 2^{2}

كـ

3 sec^{2} (x) - 4

اكتب مجددًا

3 sec^{2} (x) - 4

a = (a)^{2}

:فعْل قانون الجذور

3 = (3)^{2}

= (3)^{2} sec^{2} (x) - 4

2^{2}

كـ

4

اكتب مجددًا

= (3)^{2} sec^{2} (x) - 2^{2}

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

:فعّل قانون القوى

(3)^{2} sec^{2} (x) = (3 sec (x))^{2}

= (3 sec (x))^{2} - 2^{2}

= (3 sec (x))^{2} - 2^{2}

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

فعّل قانون فرق المربّعات

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

= (3 sec (x) + 2) (3 sec (x) - 2)

= tan (x) (3 sec (x) + 2) (3 sec (x) - 2)

tan (x) (3 sec (x) + 2) (3 sec (x) - 2) = 0

حلّ كل جزء على حدة

tan (x) = 0 or 3 sec (x) + 2 = 0 or 3 sec (x) - 2 = 0

tan (x) = 0 : x = πn

tan (x) = 0

tan (x) = 0 :

حلول عامّة لـ

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

x = 0 + πn

حلّ:

x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

3 sec (x) + 2 = 0 : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

3 sec (x) + 2 = 0

انقل

2

إلى الجانب الأيمن

3 sec (x) + 2 = 0

من الطرفين

2

اطرح

3 sec (x) + 2 - 2 = 0 - 2

بسّط

3 sec (x) = - 2

3 sec (x) = - 2

3

اقسم الطرفين على

3 sec (x) = - 2

3

اقسم الطرفين على

\frac{3 sec ( x )}{3} = \frac{- 2}{3}

بسّط

\frac{3 sec ( x )}{3} = \frac{- 2}{3}

\frac{3 sec ( x )}{3}

بسّط:

sec (x)

\frac{3 sec ( x )}{3}

3 :

إلغ العوامل المشتركة

= sec (x)

\frac{- 2}{3}

بسّط:

- \frac{2 3}{3}

\frac{- 2}{3}

\frac{- a}{b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{2}{3}

- \frac{2}{3}

حوّل لصيغة عدد كسريّ:

- \frac{2 3}{3}

- \frac{2}{3}

\frac{3}{3}

اضرب بالمرافق

= - \frac{2 3}{3 3}

33 = 3

33

a a = a

:فعْل قانون الجذور

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

sec (x) = - \frac{2 3}{3}

sec (x) = - \frac{2 3}{3}

sec (x) = - \frac{2 3}{3}

sec (x) = - \frac{2 3}{3} :

حلول عامّة لـ

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

3 sec (x) - 2 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

3 sec (x) - 2 = 0

انقل

2

إلى الجانب الأيمن

3 sec (x) - 2 = 0

للطرفين

2

أضف

3 sec (x) - 2 + 2 = 0 + 2

بسّط

3 sec (x) = 2

3 sec (x) = 2

3

اقسم الطرفين على

3 sec (x) = 2

3

اقسم الطرفين على

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

بسّط

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

\frac{3 sec ( x )}{3}

بسّط:

sec (x)

\frac{3 sec ( x )}{3}

3 :

إلغ العوامل المشتركة

= sec (x)

\frac{2}{3}

بسّط:

\frac{2 3}{3}

\frac{2}{3}

\frac{3}{3}

اضرب بالمرافق

= \frac{2 3}{3 3}

33 = 3

33

a a = a

:فعْل قانون الجذور

33 = 3

= 3

= \frac{2 3}{3}

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

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

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

sec (x) = \frac{2 3}{3} :

حلول عامّة لـ

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

وحّد الحلول

x = πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

رسم

أمثلة شائعة

csc^2(θ)=cot(θ)+1

csc^{2} (θ) = cot (θ) + 1

7cos(x)tan(x)=-8tan(x)

7 cos (x) tan (x) = - 8 tan (x)

4sin(θ)-1=2sin(θ)+1

4 sin (θ) - 1 = 2 sin (θ) + 1

3 sec^{2} (2 x) = 4

4cot(x)=cot(x)sin^2(x)

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