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شائع >

cos^2(β)-cot^2(β)=cos^2(β)cot^2(β)

الحلّ

cos^{2} (β) - cot^{2} (β) = cos^{2} (β) cot^{2} (β)

الحلّ

β = \frac{π}{2} + πn

درجات

β = 9 0^{\circ} + 18 0^{\circ} n

خطوات الحلّ

cos^{2} (β) - cot^{2} (β) = cos^{2} (β) cot^{2} (β)

من الطرفين

cos^{2} (β) cot^{2} (β)

اطرح

cos^{2} (β) - cot^{2} (β) - cos^{2} (β) cot^{2} (β) = 0

sin, cos :

عبّر بواسطة

cos^{2} (β) - cot^{2} (β) - cos^{2} (β) cot^{2} (β)

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

:Use the basic trigonometric identity

= cos^{2} (β) - (\frac{cos ( β )}{sin ( β )})^{2} - cos^{2} (β) (\frac{cos ( β )}{sin ( β )})^{2}

cos^{2} (β) - (\frac{cos ( β )}{sin ( β )})^{2} - cos^{2} (β) (\frac{cos ( β )}{sin ( β )})^{2}

بسّط:

\frac{cos ^{2} ( β ) sin ^{2} ( β ) - cos ^{2} ( β ) - cos ^{4} ( β )}{sin ^{2} ( β )}

cos^{2} (β) - (\frac{cos ( β )}{sin ( β )})^{2} - cos^{2} (β) (\frac{cos ( β )}{sin ( β )})^{2}

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

(\frac{cos ( β )}{sin ( β )})^{2}

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

:فعّل قانون القوى

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

cos^{2} (β) (\frac{cos ( β )}{sin ( β )})^{2} = \frac{cos ^{4} ( β )}{sin ^{2} ( β )}

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

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

(\frac{cos ( β )}{sin ( β )})^{2}

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

:فعّل قانون القوى

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

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

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

:اضرب كسور

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

cos^{2} (β) cos^{2} (β) = cos^{4} (β)

cos^{2} (β) cos^{2} (β)

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

cos^{2} (β) cos^{2} (β) = cos^{2 + 2} (β)

= cos^{2 + 2} (β)

2 + 2 = 4 :

اجمع الأعداد

= cos^{4} (β)

= \frac{cos ^{4} ( β )}{sin ^{2} ( β )}

= cos^{2} (β) - \frac{cos ^{2} ( β )}{sin ^{2} ( β )} - \frac{cos ^{4} ( β )}{sin ^{2} ( β )}

- \frac{cos ^{2} ( β )}{sin ^{2} ( β )} - \frac{cos ^{4} ( β )}{sin ^{2} ( β )}

وحّد الكسور:

\frac{- cos ^{2} ( β ) - cos ^{4} ( β )}{sin ^{2} ( β )}

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

فعّل القانون

= \frac{- cos ^{2} ( β ) - cos ^{4} ( β )}{sin ^{2} ( β )}

= cos^{2} (β) + \frac{- cos ^{4} ( β ) - cos ^{2} ( β )}{sin ^{2} ( β )}

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

:حوّل الأعداد لكسور

= \frac{cos ^{2} ( β ) sin ^{2} ( β )}{sin ^{2} ( β )} + \frac{- cos ^{2} ( β ) - cos ^{4} ( β )}{sin ^{2} ( β )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{cos ^{2} ( β ) sin ^{2} ( β ) - cos ^{2} ( β ) - cos ^{4} ( β )}{sin ^{2} ( β )}

= \frac{cos ^{2} ( β ) sin ^{2} ( β ) - cos ^{2} ( β ) - cos ^{4} ( β )}{sin ^{2} ( β )}

\frac{- cos ^{2} ( β ) - cos ^{4} ( β ) + cos ^{2} ( β ) sin ^{2} ( β )}{sin ^{2} ( β )} = 0

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

- cos^{2} (β) - cos^{4} (β) + cos^{2} (β) sin^{2} (β) = 0

- cos^{2} (β) - cos^{4} (β) + cos^{2} (β) sin^{2} (β)

حلل إلى عوامل:

- cos^{2} (β) (1 + cos^{2} (β) - sin^{2} (β))

- cos^{2} (β) - cos^{4} (β) + cos^{2} (β) sin^{2} (β)

a^{b + c} = a^{b} a^{c}

:فعّل قانون القوى

cos^{4} (β) = cos^{2} (β) cos^{2} (β)

= - cos^{2} (β) - cos^{2} (β) cos^{2} (β) + sin^{2} (β) cos^{2} (β)

- cos^{2} (β)

قم باخراج العامل المشترك

= - cos^{2} (β) (1 + cos^{2} (β) - sin^{2} (β))

- cos^{2} (β) (1 + cos^{2} (β) - sin^{2} (β)) = 0

حلّ كل جزء على حدة

cos^{2} (β) = 0 or 1 + cos^{2} (β) - sin^{2} (β) = 0

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

cos^{2} (β) = 0

x^{n} = 0 \Rightarrow x = 0

فعّل القانون

cos (β) = 0

cos (β) = 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}

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

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

1 + cos^{2} (β) - sin^{2} (β) = 0 : β = \frac{π}{2} + πn

1 + cos^{2} (β) - sin^{2} (β) = 0

Rewrite using trig identities

1 + cos^{2} (β) - sin^{2} (β)

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

:فعّل متطابقة الزاوية المضاعفة

= 1 + cos (2 β)

1 + cos (2 β) = 0

انقل

1

إلى الجانب الأيمن

1 + cos (2 β) = 0

من الطرفين

1

اطرح

1 + cos (2 β) - 1 = 0 - 1

بسّط

cos (2 β) = - 1

cos (2 β) = - 1

cos (2 β) = - 1 :

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}

2 β = π + 2 πn

2 β = π + 2 πn

2 β = π + 2 πn

حلّ:

β = \frac{π}{2} + πn

2 β = π + 2 πn

2

اقسم الطرفين على

2 β = π + 2 πn

2

اقسم الطرفين على

\frac{2 β}{2} = \frac{π}{2} + \frac{2 πn}{2}

بسّط

β = \frac{π}{2} + πn

β = \frac{π}{2} + πn

β = \frac{π}{2} + πn

وحّد الحلول

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

ادمج المجالات المتطابقة

β = \frac{π}{2} + πn

رسم

أمثلة شائعة

4tan(x)+4=0,0<= x<2pi

4 tan (x) + 4 = 0, 0 \leq x < 2 π

cos^3(x)=cos(x),0<= x<2pi

cos^{3} (x) = cos (x), 0 \leq x < 2 π

3-2sin^2(x)=2sin^2(x/2)

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

cot(θ)=-7/5 ,(3pi)/2 <θ<2pi,tan(θ)

cot (θ) = - \frac{7}{5}, \frac{3 π}{2} < θ < 2 π, tan (θ)

sin(r)=0.58

sin (r) = 0.58