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4cos^2(x)tan(x)-tan(x)=0

الحلّ

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

الحلّ

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

+1

درجات

x = 0^{\circ} + 18 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

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

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

حلل إلى عوامل:

tan (x) (2 cos (x) + 1) (2 cos (x) - 1)

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

tan (x)

قم باخراج العامل المشترك

= tan (x) (4 cos^{2} (x) - 1)

4 cos^{2} (x) - 1

حلل إلى عوامل:

(2 cos (x) + 1) (2 cos (x) - 1)

4 cos^{2} (x) - 1

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

كـ

4 cos^{2} (x) - 1

اكتب مجددًا

4 cos^{2} (x) - 1

2^{2}

كـ

4

اكتب مجددًا

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

1^{2}

كـ

1

اكتب مجددًا

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

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

:فعّل قانون القوى

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

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

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

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

فعّل قانون فرق المربّعات

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

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

= tan (x) (2 cos (x) + 1) (2 cos (x) - 1)

tan (x) (2 cos (x) + 1) (2 cos (x) - 1) = 0

حلّ كل جزء على حدة

tan (x) = 0 or 2 cos (x) + 1 = 0 or 2 cos (x) - 1 = 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

2 cos (x) + 1 = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

2 cos (x) + 1 = 0

انقل

1

إلى الجانب الأيمن

2 cos (x) + 1 = 0

من الطرفين

1

اطرح

2 cos (x) + 1 - 1 = 0 - 1

بسّط

2 cos (x) = - 1

2 cos (x) = - 1

2

اقسم الطرفين على

2 cos (x) = - 1

2

اقسم الطرفين على

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

بسّط

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

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

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

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}

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

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

2 cos (x) - 1 = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

2 cos (x) - 1 = 0

انقل

1

إلى الجانب الأيمن

2 cos (x) - 1 = 0

للطرفين

1

أضف

2 cos (x) - 1 + 1 = 0 + 1

بسّط

2 cos (x) = 1

2 cos (x) = 1

2

اقسم الطرفين على

2 cos (x) = 1

2

اقسم الطرفين على

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

بسّط

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

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

cos (x) = \frac{1}{2} :

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}

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

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

وحّد الحلول

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

أمثلة شائعة

sin(2x)+sin^2(x)+cos^2(x)=0

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

csc(x)= 1/(cos(x))

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

csc(θ)=(1.201)

csc (θ) = (1.201)

3cos(θ)+2=5sec(θ),0<= θ<2pi

3 cos (θ) + 2 = 5 sec (θ), 0 \leq θ < 2 π

2cos(pi/4-3x)=sqrt(2)

2 cos (\frac{π}{4} - 3 x) = 2