English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

شائع علم المثلثات >

1/(tan(x)-cot(x))=sin(x)cos(x)

الحلّ

\frac{1}{tan ( x ) - cot ( x )} = sin (x) cos (x)

الحلّ

x \in R لا يوجد حلّ لـ

خطوات الحلّ

\frac{1}{tan ( x ) - cot ( x )} = sin (x) cos (x)

من الطرفين

sin (x) cos (x)

اطرح

\frac{1}{tan ( x ) - cot ( x )} - sin (x) cos (x) = 0

\frac{1}{tan ( x ) - cot ( x )} - sin (x) cos (x)

بسّط:

\frac{1 - sin ( x ) cos ( x ) ( tan ( x ) - cot ( x ) )}{tan ( x ) - cot ( x )}

\frac{1}{tan ( x ) - cot ( x )} - sin (x) cos (x)

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

:حوّل الأعداد لكسور

= \frac{1}{tan ( x ) - cot ( x )} - \frac{sin ( x ) cos ( x ) ( tan ( x ) - cot ( x ) )}{tan ( x ) - cot ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{1 - sin ( x ) cos ( x ) ( tan ( x ) - cot ( x ) )}{tan ( x ) - cot ( x )}

\frac{1 - sin ( x ) cos ( x ) ( tan ( x ) - cot ( x ) )}{tan ( x ) - cot ( x )} = 0

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

1 - sin (x) cos (x) (tan (x) - cot (x)) = 0

sin, cos :

عبّر بواسطة

1 - (- cot (x) + tan (x)) cos (x) sin (x)

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

:Use the basic trigonometric identity

= 1 - (- \frac{cos ( x )}{sin ( x )} + tan (x)) cos (x) sin (x)

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

:Use the basic trigonometric identity

= 1 - (- \frac{cos ( x )}{sin ( x )} + \frac{sin ( x )}{cos ( x )}) cos (x) sin (x)

1 - (- \frac{cos ( x )}{sin ( x )} + \frac{sin ( x )}{cos ( x )}) cos (x) sin (x)

بسّط:

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

1 - (- \frac{cos ( x )}{sin ( x )} + \frac{sin ( x )}{cos ( x )}) cos (x) sin (x)

- \frac{cos ( x )}{sin ( x )} + \frac{sin ( x )}{cos ( x )}

وحّد:

\frac{- cos ^{2} ( x ) + sin ^{2} ( x )}{sin ( x ) cos ( x )}

- \frac{cos ( x )}{sin ( x )} + \frac{sin ( x )}{cos ( x )}

sin (x), cos (x)

المضاعف المشترك الأصغر لـ:

sin (x) cos (x)

sin (x), cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

cos (x)

= sin (x) cos (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (x) cos (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{cos ( x )}{sin ( x )} :

multiply the denominator and numerator by

cos (x)

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

For

\frac{sin ( x )}{cos ( x )} :

multiply the denominator and numerator by

sin (x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

\frac{- cos ^{2} ( x ) + sin ^{2} ( x )}{sin ( x ) cos ( x )} cos (x) sin (x)

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

:اضرب كسور

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

cos (x) :

إلغ العوامل المشتركة

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

sin (x) :

إلغ العوامل المشتركة

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

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

- (- cos^{2} (x) + sin^{2} (x)) : cos^{2} (x) - sin^{2} (x)

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

افتح أقواس

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

فعّل قوانين سالب-موجب

- (- a) = a, - (a) = - a

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

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

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

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

Rewrite using trig identities

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

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

:فعّل متطابقة الزاوية المضاعفة

= 1 + cos (2 x)

1 + cos (2 x) = 0

انقل

1

إلى الجانب الأيمن

1 + cos (2 x) = 0

من الطرفين

1

اطرح

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

بسّط

cos (2 x) = - 1

cos (2 x) = - 1

cos (2 x) = - 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 x = π + 2 πn

2 x = π + 2 πn

2 x = π + 2 πn

حلّ:

x = \frac{π}{2} + πn

2 x = π + 2 πn

2

اقسم الطرفين على

2 x = π + 2 πn

2

اقسم الطرفين على

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

بسّط

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

\frac{π}{2} + πn :

بما أنّ المعادلة غير معرّفة لـ

x \in R لا يوجد حلّ لـ

رسم

أمثلة شائعة

cot(3x)=-sqrt(3)

cot (3 x) = - 3

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

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

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

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

sqrt(7)sin(θ)-1=0

7 sin (θ) - 1 = 0

sin(3θ)=-1/(sqrt(2))

sin (3 θ) = - \frac{1}{2}