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9cos^2(x)tan(x)=4tan(x)

الحلّ

9 cos^{2} (x) tan (x) = 4 tan (x)

الحلّ

x = πn, x = 2.30052 \dots + 2 πn, x = - 2.30052 \dots + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

+1

درجات

x = 0^{\circ} + 18 0^{\circ} n, x = 131.81031 \dots^{\circ} + 36 0^{\circ} n, x = - 131.81031 \dots^{\circ} + 36 0^{\circ} n, x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n

خطوات الحلّ

9 cos^{2} (x) tan (x) = 4 tan (x)

من الطرفين

4 tan (x)

اطرح

9 cos^{2} (x) tan (x) - 4 tan (x) = 0

9 cos^{2} (x) tan (x) - 4 tan (x)

حلل إلى عوامل:

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

9 cos^{2} (x) tan (x) - 4 tan (x)

tan (x)

قم باخراج العامل المشترك

= tan (x) (9 cos^{2} (x) - 4)

9 cos^{2} (x) - 4

حلل إلى عوامل:

(3 cos (x) + 2) (3 cos (x) - 2)

9 cos^{2} (x) - 4

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

كـ

9 cos^{2} (x) - 4

اكتب مجددًا

9 cos^{2} (x) - 4

3^{2}

كـ

9

اكتب مجددًا

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

2^{2}

كـ

4

اكتب مجددًا

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

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

:فعّل قانون القوى

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

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

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

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

فعّل قانون فرق المربّعات

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

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

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

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

حلّ كل جزء على حدة

tan (x) = 0 or 3 cos (x) + 2 = 0 or 3 cos (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 cos (x) + 2 = 0 : x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

3 cos (x) + 2 = 0

انقل

2

إلى الجانب الأيمن

3 cos (x) + 2 = 0

من الطرفين

2

اطرح

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

بسّط

3 cos (x) = - 2

3 cos (x) = - 2

3

اقسم الطرفين على

3 cos (x) = - 2

3

اقسم الطرفين على

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

بسّط

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

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

Apply trig inverse properties

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

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

حلول عامّة لـ

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

3 cos (x) - 2 = 0 : x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

3 cos (x) - 2 = 0

انقل

2

إلى الجانب الأيمن

3 cos (x) - 2 = 0

للطرفين

2

أضف

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

بسّط

3 cos (x) = 2

3 cos (x) = 2

3

اقسم الطرفين على

3 cos (x) = 2

3

اقسم الطرفين على

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

بسّط

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

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

Apply trig inverse properties

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

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

حلول عامّة لـ

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

وحّد الحلول

x = πn, x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn, x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

أظهر الحلّ بالتمثيل العشريّ

x = πn, x = 2.30052 \dots + 2 πn, x = - 2.30052 \dots + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

أمثلة شائعة

3tan(x)-3=0,0<= x<= 2pi

3 tan (x) - 3 = 0, 0 \leq x \leq 2 π

(sin(57.22))/(31.5)=(sin(Q))/(24.4)

\frac{sin ( 57.2 2 ^{\circ} )}{31.5} = \frac{sin ( Q )}{24.4}

cosh (x) = 10

solvefor x,2sin^2(x)-sin(x)=1 solve for

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

arctan (x) = 180