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شائع علم المثلثات >

csc(x)-sin(x)=cot(x)*csc(x)

الحلّ

csc (x) - sin (x) = cot (x) \cdot csc (x)

الحلّ

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

درجات

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

csc (x) - sin (x) = cot (x) csc (x)

من الطرفين

cot (x) csc (x)

اطرح

csc (x) - sin (x) - cot (x) csc (x) = 0

sin, cos :

عبّر بواسطة

csc (x) - sin (x) - cot (x) csc (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

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

:اضرب كسور

= \frac{cos ( x ) \cdot 1}{sin ( x ) sin ( x )}

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

اضرب

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

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

sin (x) sin (x)

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

:فعّل قانون القوى

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

= sin^{1 + 1} (x)

1 + 1 = 2 :

اجمع الأعداد

= sin^{2} (x)

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

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

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

:حوّل الأعداد لكسور

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

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

المضاعف المشترك الأصغر لـ:

sin^{2} (x)

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

Lowest Common Multiplier (LCM)

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

= sin^{2} (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin^{2} (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

sin (x)

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

For

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

multiply the denominator and numerator by

sin^{2} (x)

\frac{sin ( x )}{1} = \frac{sin ( x ) sin ^{2} ( x )}{1 \cdot sin ^{2} ( x )} = \frac{sin ^{3} ( x )}{sin ^{2} ( x )}

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

\frac{- cos ( x ) + sin ( x ) - sin ^{3} ( x )}{sin ^{2} ( x )} = 0

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

- cos (x) + sin (x) - sin^{3} (x) = 0

a^{b} = a^{2} a^{b - 2} :

فعّل قانون القوى

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

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

بسّط:

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

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

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

وسٌع:

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

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

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

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

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

فعّل قوانين سالب-موجب

- (- a) = a

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

1 \cdot sin (x) = sin (x) :

اضرب

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

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

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

اجمع العناصر المتشابهة

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

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

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

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

حلل إلى عوامل:

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

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

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

:فعّل قانون القوى

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

= - cos (x) + cos (x) cos (x)

cos (x)

قم باخراج العامل المشترك

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

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

حلّ كل جزء على حدة

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

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

cos (x) = 0

cos (x) = 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}

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

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

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

لا يوجد حلّ

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

Rewrite using trig identities

- 1 + sin (x) cos (x)

2 sin (x) cos (x) = sin (2 x)

:فعّل متطابقة الزاوية المضاعفة

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

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

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

انقل

1

إلى الجانب الأيمن

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

للطرفين

1

أضف

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

بسّط

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

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

2

اضرب الطرفين بـ

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

2

اضرب الطرفين بـ

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

بسّط

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

لا يوجد حلّ

وحّد الحلول

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

رسم

أمثلة شائعة

sin(x)= 4/3

sin (x) = \frac{4}{3}

1=sin(t)+sqrt(3)cos(t)

1 = sin (t) + 3 cos (t)

cosh(x)= 5/4

cosh (x) = \frac{5}{4}

sec(θ)-sqrt(2)tan(θ)=0

sec (θ) - 2 tan (θ) = 0

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

2 sin (x - \frac{π}{3}) = - 2