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شائع علم المثلثات >

3(cos^2(a))/(sin^2(a))=(cot(a))^2

الحلّ

3 \frac{cos ^{2} ( a )}{sin ^{2} ( a )} = (cot (a))^{2}

الحلّ

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

درجات

a = 9 0^{\circ} + 36 0^{\circ} n, a = 27 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

3 \cdot \frac{cos ^{2} ( a )}{sin ^{2} ( a )} = (cot (a))^{2}

من الطرفين

(cot (a))^{2}

اطرح

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

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

بسّط:

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

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

cot^{2} (a) = \frac{cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

:حوّل الأعداد لكسور

= \frac{3 cos ^{2} ( a )}{sin ^{2} ( a )} - \frac{cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{3 cos ^{2} ( a ) - cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

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

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

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

3 cos^{2} (a) - cot^{2} (a) sin^{2} (a)

حلل إلى عوامل:

(3 cos (a) + cot (a) sin (a)) (3 cos (a) - cot (a) sin (a))

3 cos^{2} (a) - cot^{2} (a) sin^{2} (a)

(cot (a) sin (a))^{2}

كـ

cot^{2} (a) sin^{2} (a)

اكتب مجددًا

cot^{2} (a) sin^{2} (a)

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

:فعّل قانون القوى

cot^{2} (a) sin^{2} (a) = (cot (a) sin (a))^{2}

= (cot (a) sin (a))^{2}

= 3 cos^{2} (a) - (cot (a) sin (a))^{2}

(3 cos (a))^{2} - (cot (a) sin (a))^{2}

كـ

3 cos^{2} (a) - (cot (a) sin (a))^{2}

اكتب مجددًا

3 cos^{2} (a) - (cot (a) sin (a))^{2}

a = (a)^{2}

:فعْل قانون الجذور

3 = (3)^{2}

= (3)^{2} cos^{2} (a) - (cot (a) sin (a))^{2}

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

:فعّل قانون القوى

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

= (3 cos (a))^{2} - (cot (a) sin (a))^{2}

= (3 cos (a))^{2} - (cot (a) sin (a))^{2}

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

فعّل قانون فرق المربّعات

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

= (3 cos (a) + cot (a) sin (a)) (3 cos (a) - cot (a) sin (a))

(3 cos (a) + cot (a) sin (a)) (3 cos (a) - cot (a) sin (a)) = 0

حلّ كل جزء على حدة

3 cos (a) + cot (a) sin (a) = 0 or 3 cos (a) - cot (a) sin (a) = 0

3 cos (a) + cot (a) sin (a) = 0 : a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

3 cos (a) + cot (a) sin (a) = 0

Rewrite using trig identities

cos (a) 3 + cot (a) sin (a)

cot (x) = \frac{cos ( x )}{sin ( x )}

:Use the basic trigonometric identity

= cos (a) 3 + \frac{cos ( a )}{sin ( a )} sin (a)

\frac{cos ( a )}{sin ( a )} sin (a) = cos (a)

\frac{cos ( a )}{sin ( a )} sin (a)

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

:اضرب كسور

= \frac{cos ( a ) sin ( a )}{sin ( a )}

sin (a) :

إلغ العوامل المشتركة

= cos (a)

= 3 cos (a) + cos (a)

cos (a) + cos (a) 3 = 0

بالاستعانة بطريقة التعويض

cos (a) + cos (a) 3 = 0

cos (a) = u :

على افتراض أنّ

u + u 3 = 0

u + u 3 = 0 : u = 0

u + u 3 = 0

u + u 3

حلل إلى عوامل:

(1 + 3) u

u + u 3

u

قم باخراج العامل المشترك

= u (1 + 3)

(1 + 3) u = 0

1 + 3

اقسم الطرفين على

(1 + 3) u = 0

1 + 3

اقسم الطرفين على

\frac{( 1 + 3 ) u}{1 + 3} = \frac{0}{1 + 3}

بسّط

u = 0

u = 0

u = cos (a)

استبدل مجددًا

cos (a) = 0

cos (a) = 0

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

cos (a) = 0

cos (a) = 0 :

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}

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

وحّد الحلول

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

3 cos (a) - cot (a) sin (a) = 0 : a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

3 cos (a) - cot (a) sin (a) = 0

Rewrite using trig identities

cos (a) 3 - cot (a) sin (a)

cot (x) = \frac{cos ( x )}{sin ( x )}

:Use the basic trigonometric identity

= cos (a) 3 - \frac{cos ( a )}{sin ( a )} sin (a)

\frac{cos ( a )}{sin ( a )} sin (a) = cos (a)

\frac{cos ( a )}{sin ( a )} sin (a)

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

:اضرب كسور

= \frac{cos ( a ) sin ( a )}{sin ( a )}

sin (a) :

إلغ العوامل المشتركة

= cos (a)

= 3 cos (a) - cos (a)

- cos (a) + cos (a) 3 = 0

بالاستعانة بطريقة التعويض

- cos (a) + cos (a) 3 = 0

cos (a) = u :

على افتراض أنّ

- u + u 3 = 0

- u + u 3 = 0 : u = 0

- u + u 3 = 0

- u + u 3

حلل إلى عوامل:

(- 1 + 3) u

- u + u 3

u

قم باخراج العامل المشترك

= u (- 1 + 3)

(- 1 + 3) u = 0

- 1 + 3

اقسم الطرفين على

(- 1 + 3) u = 0

- 1 + 3

اقسم الطرفين على

\frac{( - 1 + 3 ) u}{- 1 + 3} = \frac{0}{- 1 + 3}

بسّط

u = 0

u = 0

u = cos (a)

استبدل مجددًا

cos (a) = 0

cos (a) = 0

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

cos (a) = 0

cos (a) = 0 :

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}

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

وحّد الحلول

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

وحّد الحلول

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

رسم

أمثلة شائعة

cos(x)=(-2)/(sqrt(2))

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

tan(x)=-0.22

tan (x) = - 0.22

solvefor θ,1+cos(θ)=3cos(θ)

so l v e f or θ, 1 + cos (θ) = 3 cos (θ)

3sin(x)-4cos(x)=0

3 sin (x) - 4 cos (x) = 0

sin(θ)= 1/4 ,0<θ<90

sin (θ) = \frac{1}{4}, 0^{\circ} < θ < 9 0^{\circ}