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شائع علم المثلثات >

1/(cot(x))-(sec(x))/(csc(x))=cos(x)

الحلّ

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} = cos (x)

الحلّ

x \in R لا يوجد حلّ لـ

خطوات الحلّ

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} = cos (x)

من الطرفين

cos (x)

اطرح

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} - cos (x) = 0

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} - cos (x)

بسّط:

\frac{csc ( x ) - sec ( x ) cot ( x ) - cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )}

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} - cos (x)

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

:حوّل الأعداد لكسور

= \frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} - \frac{cos ( x )}{1}

cot (x), csc (x), 1

المضاعف المشترك الأصغر لـ:

cot (x) csc (x)

cot (x), csc (x), 1

Lowest Common Multiplier (LCM)

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

= cot (x) csc (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

cot (x) csc (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

csc (x)

\frac{1}{cot ( x )} = \frac{1 \cdot csc ( x )}{cot ( x ) csc ( x )} = \frac{csc ( x )}{cot ( x ) csc ( x )}

For

\frac{sec ( x )}{csc ( x )} :

multiply the denominator and numerator by

cot (x)

\frac{sec ( x )}{csc ( x )} = \frac{sec ( x ) cot ( x )}{csc ( x ) cot ( x )}

For

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

multiply the denominator and numerator by

cot (x) csc (x)

\frac{cos ( x )}{1} = \frac{cos ( x ) cot ( x ) csc ( x )}{1 \cdot cot ( x ) csc ( x )} = \frac{cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )}

= \frac{csc ( x )}{cot ( x ) csc ( x )} - \frac{sec ( x ) cot ( x )}{csc ( x ) cot ( x )} - \frac{cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{csc ( x ) - sec ( x ) cot ( x ) - cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )}

\frac{csc ( x ) - sec ( x ) cot ( x ) - cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )} = 0

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

csc (x) - sec (x) cot (x) - cos (x) cot (x) csc (x) = 0

sin, cos :

عبّر بواسطة

csc (x) - cot (x) sec (x) - cos (x) cot (x) csc (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

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

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

:اضرب كسور

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

cos (x) :

إلغ العوامل المشتركة

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

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

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

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

:اضرب كسور

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

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

cos (x) \cdot 1 \cdot cos (x)

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

:فعّل قانون القوى

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

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

بسّط

= cos^{2} (x)

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

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

1 \cdot \frac{1}{sin ( x )} - 1 \cdot \frac{1}{sin ( x )} = 0 :

اجمع العناصر المتشابهة

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

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

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

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

- cos^{2} (x) = 0

- 1

اقسم الطرفين على

- cos^{2} (x) = 0

- 1

اقسم الطرفين على

- cos^{2} (x) = 0

- 1

اقسم الطرفين على

\frac{- cos ^{2} ( x )}{- 1} = \frac{0}{- 1}

بسّط

cos^{2} (x) = 0

cos^{2} (x) = 0

x^{n} = 0 \Rightarrow x = 0

فعّل القانون

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

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

بما أنّ المعادلة غير معرّفة لـ

x \in R لا يوجد حلّ لـ

رسم

أمثلة شائعة

sin(x)=1,2pi<= x<= 4pi

sin (x) = 1, 2 π \leq x \leq 4 π

cos(a)=1

cos (a) = 1

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

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

sec(θ)= 7/5

sec (θ) = \frac{7}{5}

(sin(x))/(cos(x))=0

\frac{sin ( x )}{cos ( x )} = 0