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شائع علم المثلثات >

(sec(B)+csc(B))/(1+tan(B))=csc^2(B)

الحلّ

\frac{sec ( B ) + csc ( B )}{1 + tan ( B )} = csc^{2} (B)

الحلّ

B \in R لا يوجد حلّ لـ

خطوات الحلّ

\frac{sec ( B ) + csc ( B )}{1 + tan ( B )} = csc^{2} (B)

من الطرفين

csc^{2} (B)

اطرح

\frac{sec ( B ) + csc ( B )}{1 + tan ( B )} - csc^{2} (B) = 0

\frac{sec ( B ) + csc ( B )}{1 + tan ( B )} - csc^{2} (B)

بسّط:

\frac{sec ( B ) + csc ( B ) - csc ^{2} ( B ) ( 1 + tan ( B ) )}{1 + tan ( B )}

\frac{sec ( B ) + csc ( B )}{1 + tan ( B )} - csc^{2} (B)

csc^{2} (B) = \frac{csc ^{2} ( B ) ( 1 + tan ( B ) )}{1 + tan ( B )}

:حوّل الأعداد لكسور

= \frac{sec ( B ) + csc ( B )}{1 + tan ( B )} - \frac{csc ^{2} ( B ) ( 1 + tan ( B ) )}{1 + tan ( B )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{sec ( B ) + csc ( B ) - csc ^{2} ( B ) ( 1 + tan ( B ) )}{1 + tan ( B )}

\frac{sec ( B ) + csc ( B ) - csc ^{2} ( B ) ( 1 + tan ( B ) )}{1 + tan ( B )} = 0

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

sec (B) + csc (B) - csc^{2} (B) (1 + tan (B)) = 0

sin, cos :

عبّر بواسطة

csc (B) + sec (B) - (1 + tan (B)) csc^{2} (B)

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

:Use the basic trigonometric identity

= \frac{1}{sin ( B )} + sec (B) - (1 + tan (B)) (\frac{1}{sin ( B )})^{2}

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

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

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

وحّد:

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

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

1 = \frac{1 cos ( B )}{cos ( B )}

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

1 \cdot cos (B) = cos (B) :

اضرب

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

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

(\frac{1}{sin ( B )})^{2} = \frac{1}{sin ^{2} ( B )}

(\frac{1}{sin ( B )})^{2}

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

:فعّل قانون القوى

= \frac{1 ^{2}}{sin ^{2} ( B )}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{sin ^{2} ( B )}

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

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

:اضرب كسور

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

(cos (B) + sin (B)) \cdot 1 = cos (B) + sin (B)

(cos (B) + sin (B)) \cdot 1

(cos (B) + sin (B)) \cdot 1 = (cos (B) + sin (B)) :

اضرب

= (cos (B) + sin (B))

(a) = a

:احذف الأقواس

= cos (B) + sin (B)

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

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

sin (B), cos (B), cos (B) sin^{2} (B)

المضاعف المشترك الأصغر لـ:

sin^{2} (B) cos (B)

sin (B), cos (B), cos (B) sin^{2} (B)

Lowest Common Multiplier (LCM)

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

= sin^{2} (B) cos (B)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin^{2} (B) cos (B)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{1}{sin ( B )} :

multiply the denominator and numerator by

sin (B) cos (B)

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

For

\frac{1}{cos ( B )} :

multiply the denominator and numerator by

sin^{2} (B)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

- (cos (B) + sin (B)) : - cos (B) - sin (B)

- (cos (B) + sin (B))

افتح أقواس

= - (cos (B)) - (sin (B))

فعّل قوانين سالب-موجب

+ (- a) = - a

= - cos (B) - sin (B)

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

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

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

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

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

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

حلل إلى عوامل:

(sin (B) + cos (B)) (sin (B) - 1)

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

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

sin (B) (sin (B) + cos (B))

sin^{2} (B) + sin (B) cos (B)

من

sin (B)

اخرج العامل

sin^{2} (B) + sin (B) cos (B)

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

:فعّل قانون القوى

sin^{2} (B) = sin (B) sin (B)

= sin (B) sin (B) + sin (B) cos (B)

sin (B)

قم باخراج العامل المشترك

= sin (B) (sin (B) + cos (B))

- (sin (B) + cos (B))

- sin (B) - cos (B)

من

- 1

اخرج العامل

- sin (B) - cos (B)

- 1

قم باخراج العامل المشترك

= - (sin (B) + cos (B))

= sin (B) (sin (B) + cos (B)) - (sin (B) + cos (B))

sin (B) + cos (B)

قم باخراج العامل المشترك

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

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

حلّ كل جزء على حدة

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

sin (B) + cos (B) = 0 : B = \frac{3 π}{4} + πn

sin (B) + cos (B) = 0

Rewrite using trig identities

sin (B) + cos (B) = 0

cos (B) \neq = 0, cos (B)

اقسم الطرفين على

\frac{sin ( B ) + cos ( B )}{cos ( B )} = \frac{0}{cos ( B )}

بسّط

\frac{sin ( B )}{cos ( B )} + 1 = 0

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

:Use the basic trigonometric identity

tan (B) + 1 = 0

tan (B) + 1 = 0

انقل

1

إلى الجانب الأيمن

tan (B) + 1 = 0

من الطرفين

1

اطرح

tan (B) + 1 - 1 = 0 - 1

بسّط

tan (B) = - 1

tan (B) = - 1

tan (B) = - 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}

B = \frac{3 π}{4} + πn

B = \frac{3 π}{4} + πn

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

sin (B) - 1 = 0

انقل

1

إلى الجانب الأيمن

sin (B) - 1 = 0

للطرفين

1

أضف

sin (B) - 1 + 1 = 0 + 1

بسّط

sin (B) = 1

sin (B) = 1

sin (B) = 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}

B = \frac{π}{2} + 2 πn

B = \frac{π}{2} + 2 πn

وحّد الحلول

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

\frac{3 π}{4} + πn, \frac{π}{2} + 2 πn :

بما أنّ المعادلة غير معرّفة لـ

B \in R لا يوجد حلّ لـ

رسم

أمثلة شائعة

4=sqrt(2)csc(3x)

4 = 2 csc (3 x)

21=24+8cos((pix)/6)

21 = 24 + 8 cos (\frac{π x}{6})

sin(x)=(16sin(31))/(12)

sin (x) = \frac{16 sin ( 3 1 ^{\circ} )}{12}

sinh(z)-cosh(z)=0

sinh (z) - cosh (z) = 0

cos(x)=-1.588908648

cos (x) = - 1.588908648