English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

شائع علم المثلثات >

(1+csc(γ))/(cot(γ)+cos(γ))=csc(γ)

الحلّ

\frac{1 + csc ( γ )}{cot ( γ ) + cos ( γ )} = csc (γ)

الحلّ

γ = \frac{π}{4} + πn

درجات

γ = 4 5^{\circ} + 18 0^{\circ} n

خطوات الحلّ

\frac{1 + csc ( γ )}{cot ( γ ) + cos ( γ )} = csc (γ)

من الطرفين

csc (γ)

اطرح

\frac{1 + csc ( γ )}{cot ( γ ) + cos ( γ )} - csc (γ) = 0

\frac{1 + csc ( γ )}{cot ( γ ) + cos ( γ )} - csc (γ)

بسّط:

\frac{1 + csc ( γ ) - csc ( γ ) ( cot ( γ ) + cos ( γ ) )}{cot ( γ ) + cos ( γ )}

\frac{1 + csc ( γ )}{cot ( γ ) + cos ( γ )} - csc (γ)

csc (γ) = \frac{csc ( γ ) ( cot ( γ ) + cos ( γ ) )}{cot ( γ ) + cos ( γ )}

:حوّل الأعداد لكسور

= \frac{1 + csc ( γ )}{cot ( γ ) + cos ( γ )} - \frac{csc ( γ ) ( cot ( γ ) + cos ( γ ) )}{cot ( γ ) + cos ( γ )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{1 + csc ( γ ) - csc ( γ ) ( cot ( γ ) + cos ( γ ) )}{cot ( γ ) + cos ( γ )}

\frac{1 + csc ( γ ) - csc ( γ ) ( cot ( γ ) + cos ( γ ) )}{cot ( γ ) + cos ( γ )} = 0

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

1 + csc (γ) - csc (γ) (cot (γ) + cos (γ)) = 0

sin, cos :

عبّر بواسطة

1 + csc (γ) - (cos (γ) + cot (γ)) csc (γ)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

(cos (γ) + \frac{cos ( γ )}{sin ( γ )}) \frac{1}{sin ( γ )} = \frac{cos ( γ ) sin ( γ ) + cos ( γ )}{sin ^{2} ( γ )}

(cos (γ) + \frac{cos ( γ )}{sin ( γ )}) \frac{1}{sin ( γ )}

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

:اضرب كسور

= \frac{1 \cdot ( cos ( γ ) + \frac{c o s ( γ )}{s i n ( γ )} )}{sin ( γ )}

1 \cdot (cos (γ) + \frac{cos ( γ )}{sin ( γ )}) = cos (γ) + \frac{cos ( γ )}{sin ( γ )}

1 \cdot (cos (γ) + \frac{cos ( γ )}{sin ( γ )})

1 \cdot (cos (γ) + \frac{cos ( γ )}{sin ( γ )}) = (cos (γ) + \frac{cos ( γ )}{sin ( γ )}) :

اضرب

= (cos (γ) + \frac{cos ( γ )}{sin ( γ )})

(a) = a

:احذف الأقواس

= cos (γ) + \frac{cos ( γ )}{sin ( γ )}

= \frac{cos ( γ ) + \frac{c o s ( γ )}{s i n ( γ )}}{sin ( γ )}

cos (γ) + \frac{cos ( γ )}{sin ( γ )}

وحّد:

\frac{cos ( γ ) sin ( γ ) + cos ( γ )}{sin ( γ )}

cos (γ) + \frac{cos ( γ )}{sin ( γ )}

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

:حوّل الأعداد لكسور

= \frac{cos ( γ ) sin ( γ )}{sin ( γ )} + \frac{cos ( γ )}{sin ( γ )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{cos ( γ ) sin ( γ ) + cos ( γ )}{sin ( γ )}

= \frac{\frac{c o s ( γ ) s i n ( γ ) + c o s ( γ )}{s i n ( γ )}}{sin ( γ )}

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

: استخدم ميزات الكسور التالية

= \frac{cos ( γ ) sin ( γ ) + cos ( γ )}{sin ( γ ) sin ( γ )}

sin (γ) sin (γ) = sin^{2} (γ)

sin (γ) sin (γ)

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

:فعّل قانون القوى

sin (γ) sin (γ) = sin^{1 + 1} (γ)

= sin^{1 + 1} (γ)

1 + 1 = 2 :

اجمع الأعداد

= sin^{2} (γ)

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

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

1 = \frac{1}{1}

:حوّل الأعداد لكسور

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

1, sin (γ), sin^{2} (γ)

المضاعف المشترك الأصغر لـ:

sin^{2} (γ)

1, sin (γ), sin^{2} (γ)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear in at least one of the factored expressions

= sin^{2} (γ)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin^{2} (γ)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{1}{1} :

multiply the denominator and numerator by

sin^{2} (γ)

\frac{1}{1} = \frac{1 \cdot sin ^{2} ( γ )}{1 \cdot sin ^{2} ( γ )} = \frac{sin ^{2} ( γ )}{sin ^{2} ( γ )}

For

\frac{1}{sin ( γ )} :

multiply the denominator and numerator by

sin (γ)

\frac{1}{sin ( γ )} = \frac{1 \cdot sin ( γ )}{sin ( γ ) sin ( γ )} = \frac{sin ( γ )}{sin ^{2} ( γ )}

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

- (cos (γ) sin (γ) + cos (γ)) : - cos (γ) sin (γ) - cos (γ)

- (cos (γ) sin (γ) + cos (γ))

افتح أقواس

= - (cos (γ) sin (γ)) - (cos (γ))

فعّل قوانين سالب-موجب

+ (- a) = - a

= - cos (γ) sin (γ) - cos (γ)

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

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

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

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

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

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

حلل إلى عوامل:

(1 + sin (γ)) (- cos (γ) + sin (γ))

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

cos (γ)

قم باخراج العامل المشترك

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

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

:فعّل قانون القوى

sin^{2} (γ) = sin (γ) sin (γ)

= - cos (γ) (1 + sin (γ)) + sin (γ) + sin (γ) sin (γ)

sin (γ)

قم باخراج العامل المشترك

= - cos (γ) (1 + sin (γ)) + sin (γ) (1 + sin (γ))

(1 + sin (γ))

قم باخراج العامل المشترك

= (1 + sin (γ)) (- cos (γ) + sin (γ))

(1 + sin (γ)) (- cos (γ) + sin (γ)) = 0

حلّ كل جزء على حدة

1 + sin (γ) = 0 or - cos (γ) + sin (γ) = 0

1 + sin (γ) = 0 : γ = \frac{3 π}{2} + 2 πn

1 + sin (γ) = 0

انقل

1

إلى الجانب الأيمن

1 + sin (γ) = 0

من الطرفين

1

اطرح

1 + sin (γ) - 1 = 0 - 1

بسّط

sin (γ) = - 1

sin (γ) = - 1

sin (γ) = - 1 :

حلول عامّة لـ

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

γ = \frac{3 π}{2} + 2 πn

γ = \frac{3 π}{2} + 2 πn

- cos (γ) + sin (γ) = 0 : γ = \frac{π}{4} + πn

- cos (γ) + sin (γ) = 0

Rewrite using trig identities

- cos (γ) + sin (γ) = 0

cos (γ) \neq = 0, cos (γ)

اقسم الطرفين على

\frac{- cos ( γ ) + sin ( γ )}{cos ( γ )} = \frac{0}{cos ( γ )}

بسّط

- 1 + \frac{sin ( γ )}{cos ( γ )} = 0

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

:Use the basic trigonometric identity

- 1 + tan (γ) = 0

- 1 + tan (γ) = 0

انقل

1

إلى الجانب الأيمن

- 1 + tan (γ) = 0

للطرفين

1

أضف

- 1 + tan (γ) + 1 = 0 + 1

بسّط

tan (γ) = 1

tan (γ) = 1

tan (γ) = 1 :

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}

γ = \frac{π}{4} + πn

γ = \frac{π}{4} + πn

وحّد الحلول

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

\frac{3 π}{2} + 2 πn :

بما أنّ المعادلة غير معرّفة لـ

γ = \frac{π}{4} + πn

رسم

أمثلة شائعة

sin(θ)= 2/6

sin (θ) = \frac{2}{6}

3sin(2x)=-2

3 sin (2 x) = - 2

solvefor x,arctan(y)=2arctan(x)

so l v e f or x, arctan (y) = 2 arctan (x)

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

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

tan(x)=(7.3)/(6.8)

tan (x) = \frac{7.3}{6.8}