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شائع علم المثلثات >

(cot^2(x))/(1+sin(x))=(csc(x)-1)/(sin^2(x))

الحلّ

\frac{cot ^{2} ( x )}{1 + sin ( x )} = \frac{csc ( x ) - 1}{sin ^{2} ( x )}

الحلّ

x = \frac{π}{2} + 2 πn

درجات

x = 9 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

\frac{cot ^{2} ( x )}{1 + sin ( x )} = \frac{csc ( x ) - 1}{sin ^{2} ( x )}

من الطرفين

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

اطرح

\frac{cot ^{2} ( x )}{1 + sin ( x )} - \frac{csc ( x ) - 1}{sin ^{2} ( x )} = 0

\frac{cot ^{2} ( x )}{1 + sin ( x )} - \frac{csc ( x ) - 1}{sin ^{2} ( x )}

بسّط:

\frac{cot ^{2} ( x ) sin ^{2} ( x ) - ( csc ( x ) - 1 ) ( sin ( x ) + 1 )}{sin ^{2} ( x ) ( sin ( x ) + 1 )}

\frac{cot ^{2} ( x )}{1 + sin ( x )} - \frac{csc ( x ) - 1}{sin ^{2} ( x )}

1 + sin (x), sin^{2} (x)

المضاعف المشترك الأصغر لـ:

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

1 + sin (x), sin^{2} (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

1 + sin (x)

sin^{2} (x)

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

اكتب مجددًا الكسور بحيث يكون المقام مشترك

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

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{cot ^{2} ( x )}{1 + sin ( x )} :

multiply the denominator and numerator by

sin^{2} (x)

\frac{cot ^{2} ( x )}{1 + sin ( x )} = \frac{cot ^{2} ( x ) sin ^{2} ( x )}{( 1 + sin ( x ) ) sin ^{2} ( x )}

For

\frac{csc ( x ) - 1}{sin ^{2} ( x )} :

multiply the denominator and numerator by

sin (x) + 1

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

= \frac{cot ^{2} ( x ) sin ^{2} ( x )}{( 1 + sin ( x ) ) sin ^{2} ( x )} - \frac{( csc ( x ) - 1 ) ( sin ( x ) + 1 )}{sin ^{2} ( x ) ( sin ( x ) + 1 )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{cot ^{2} ( x ) sin ^{2} ( x ) - ( csc ( x ) - 1 ) ( sin ( x ) + 1 )}{sin ^{2} ( x ) ( sin ( x ) + 1 )}

\frac{cot ^{2} ( x ) sin ^{2} ( x ) - ( csc ( x ) - 1 ) ( sin ( x ) + 1 )}{sin ^{2} ( x ) ( sin ( x ) + 1 )} = 0

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

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

Rewrite using trig identities

- (- 1 + csc (x)) (1 + sin (x)) + cot^{2} (x) sin^{2} (x)

1 + cot^{2} (x) = csc^{2} (x)

:فعّل نطريّة فيتاغوروس

cot^{2} (x) = csc^{2} (x) - 1

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

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

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

حلل إلى عوامل:

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

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

- 1 + csc^{2} (x)

حلل إلى عوامل:

(csc (x) + 1) (csc (x) - 1)

- 1 + csc^{2} (x)

1^{2}

كـ

1

اكتب مجددًا

= csc^{2} (x) - 1^{2}

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

فعّل قانون فرق المربّعات

csc^{2} (x) - 1^{2} = (csc (x) + 1) (csc (x) - 1)

= (csc (x) + 1) (csc (x) - 1)

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

(- 1 + csc (x))

قم باخراج العامل المشترك

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

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

وسٌع:

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

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

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

- (1 + sin (x)) : - 1 - sin (x)

- (1 + sin (x))

افتح أقواس

= - (1) - (sin (x))

فعّل قوانين سالب-موجب

+ (- a) = - a

= - 1 - sin (x)

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

sin^{2} (x) (1 + csc (x))

وسٌع:

sin^{2} (x) + sin^{2} (x) csc (x)

sin^{2} (x) (1 + csc (x))

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = sin^{2} (x), b = 1, c = csc (x)

= sin^{2} (x) \cdot 1 + sin^{2} (x) csc (x)

= 1 \cdot sin^{2} (x) + sin^{2} (x) csc (x)

1 \cdot sin^{2} (x) = sin^{2} (x) :

اضرب

= sin^{2} (x) + sin^{2} (x) csc (x)

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

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

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

حلّ كل جزء على حدة

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

- 1 + csc (x) = 0 : x = \frac{π}{2} + 2 πn

- 1 + csc (x) = 0

انقل

1

إلى الجانب الأيمن

- 1 + csc (x) = 0

للطرفين

1

أضف

- 1 + csc (x) + 1 = 0 + 1

بسّط

csc (x) = 1

csc (x) = 1

csc (x) = 1 :

حلول عامّة لـ

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

- 1 - sin (x) + sin^{2} (x) + sin^{2} (x) csc (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

Rewrite using trig identities

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

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

:Use the basic trigonometric identity

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

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

بسّط:

sin^{2} (x) - 1

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

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

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

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

:اضرب كسور

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

1 \cdot sin^{2} (x) = sin^{2} (x) :

اضرب

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

sin (x) :

إلغ العوامل المشتركة

= sin (x)

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

جمّع التعابير المتشابهة

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

- sin (x) + sin (x) = 0 :

اجمع العناصر المتشابهة

= sin^{2} (x) - 1

= sin^{2} (x) - 1

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

بالاستعانة بطريقة التعويض

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

sin (x) = u :

على افتراض أنّ

- 1 + u^{2} = 0

- 1 + u^{2} = 0 : u = 1, u = - 1

- 1 + u^{2} = 0

انقل

1

إلى الجانب الأيمن

- 1 + u^{2} = 0

للطرفين

1

أضف

- 1 + u^{2} + 1 = 0 + 1

بسّط

u^{2} = 1

u^{2} = 1

x = f (a), - f (a)

الحلول هي

x^{2} = f (a)

لـ

u = 1, u = - 1

1 = 1

1

1 = 1

فعّل القانون

= 1

- 1 = - 1

- 1

1 = 1

فعّل القانون

= - 1

u = 1, u = - 1

u = sin (x)

استبدل مجددًا

sin (x) = 1, sin (x) = - 1

sin (x) = 1, sin (x) = - 1

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

sin (x) = 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}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

sin (x) = - 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}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

وحّد الحلول

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

وحّد الحلول

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

\frac{3 π}{2} + 2 πn :

بما أنّ المعادلة غير معرّفة لـ

x = \frac{π}{2} + 2 πn

رسم

أمثلة شائعة

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tan (x) = 2.3

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

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