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شائع علم المثلثات >

tan(2θ-10)=cot(θ)

الحلّ

tan (2 θ - 1 0^{\circ}) = cot (θ)

الحلّ

θ = 33.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}, θ = 93.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

راديان

θ = \frac{5 π}{27} + \frac{2 π}{3} n, θ = \frac{14 π}{27} + \frac{2 π}{3} n

خطوات الحلّ

tan (2 θ - 1 0^{\circ}) = cot (θ)

من الطرفين

cot (θ)

اطرح

tan (2 θ - 1 0^{\circ}) - cot (θ) = 0

sin, cos :

عبّر بواسطة

- cot (θ) + tan (- 1 0^{\circ} + 2 θ)

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

:Use the basic trigonometric identity

= - \frac{cos ( θ )}{sin ( θ )} + tan (- 1 0^{\circ} + 2 θ)

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

:Use the basic trigonometric identity

= - \frac{cos ( θ )}{sin ( θ )} + \frac{sin ( - 1 0 ^{\circ} + 2 θ )}{cos ( - 1 0 ^{\circ} + 2 θ )}

- \frac{cos ( θ )}{sin ( θ )} + \frac{sin ( - 1 0 ^{\circ} + 2 θ )}{cos ( - 1 0 ^{\circ} + 2 θ )}

بسّط:

\frac{- cos ( θ ) cos ( \frac{36 θ - 18 0 ^{\circ}}{18} ) + sin ( \frac{- 18 0 ^{\circ} + 36 θ}{18} ) sin ( θ )}{sin ( θ ) cos ( \frac{36 θ - 18 0 ^{\circ}}{18} )}

- \frac{cos ( θ )}{sin ( θ )} + \frac{sin ( - 1 0 ^{\circ} + 2 θ )}{cos ( - 1 0 ^{\circ} + 2 θ )}

\frac{sin ( - 1 0 ^{\circ} + 2 θ )}{cos ( - 1 0 ^{\circ} + 2 θ )} = \frac{sin ( \frac{- 18 0 ^{\circ} + 36 θ}{18} )}{cos ( \frac{- 18 0 ^{\circ} + 36 θ}{18} )}

\frac{sin ( - 1 0 ^{\circ} + 2 θ )}{cos ( - 1 0 ^{\circ} + 2 θ )}

- 1 0^{\circ} + 2 θ

وحّد:

\frac{- 18 0 ^{\circ} + 36 θ}{18}

- 1 0^{\circ} + 2 θ

2 θ = \frac{2 θ 18}{18}

:حوّل الأعداد لكسور

= - 1 0^{\circ} + \frac{2 θ \cdot 18}{18}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{- 18 0 ^{\circ} + 2 θ \cdot 18}{18}

2 \cdot 18 = 36 :

اضرب الأعداد

= \frac{- 18 0 ^{\circ} + 36 θ}{18}

= \frac{sin ( - 1 0 ^{\circ} + 2 θ )}{cos ( \frac{- 18 0 ^{\circ} + 36 θ}{18} )}

- 1 0^{\circ} + 2 θ

وحّد:

\frac{- 18 0 ^{\circ} + 36 θ}{18}

- 1 0^{\circ} + 2 θ

2 θ = \frac{2 θ 18}{18}

:حوّل الأعداد لكسور

= - 1 0^{\circ} + \frac{2 θ \cdot 18}{18}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{- 18 0 ^{\circ} + 2 θ \cdot 18}{18}

2 \cdot 18 = 36 :

اضرب الأعداد

= \frac{- 18 0 ^{\circ} + 36 θ}{18}

= \frac{sin ( \frac{- 18 0 ^{\circ} + 36 θ}{18} )}{cos ( \frac{- 18 0 ^{\circ} + 36 θ}{18} )}

= - \frac{cos ( θ )}{sin ( θ )} + \frac{sin ( \frac{36 θ - 18 0 ^{\circ}}{18} )}{cos ( \frac{36 θ - 18 0 ^{\circ}}{18} )}

sin (θ), cos (\frac{- 18 0 ^{\circ} + 36 θ}{18})

المضاعف المشترك الأصغر لـ:

sin (θ) cos (\frac{36 θ - 18 0 ^{\circ}}{18})

sin (θ), cos (\frac{- 18 0 ^{\circ} + 36 θ}{18})

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (θ)

cos (\frac{- 18 0 ^{\circ} + 36 θ}{18})

= sin (θ) cos (\frac{36 θ - 18 0 ^{\circ}}{18})

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (θ) cos (\frac{36 θ - 18 0 ^{\circ}}{18})

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{cos ( θ )}{sin ( θ )} :

multiply the denominator and numerator by

cos (\frac{36 θ - 18 0 ^{\circ}}{18})

\frac{cos ( θ )}{sin ( θ )} = \frac{cos ( θ ) cos ( \frac{36 θ - 18 0 ^{\circ}}{18} )}{sin ( θ ) cos ( \frac{36 θ - 18 0 ^{\circ}}{18} )}

For

\frac{sin ( \frac{- 18 0 ^{\circ} + 36 θ}{18} )}{cos ( \frac{- 18 0 ^{\circ} + 36 θ}{18} )} :

multiply the denominator and numerator by

sin (θ)

\frac{sin ( \frac{- 18 0 ^{\circ} + 36 θ}{18} )}{cos ( \frac{- 18 0 ^{\circ} + 36 θ}{18} )} = \frac{sin ( \frac{- 18 0 ^{\circ} + 36 θ}{18} ) sin ( θ )}{cos ( \frac{- 18 0 ^{\circ} + 36 θ}{18} ) sin ( θ )}

= - \frac{cos ( θ ) cos ( \frac{36 θ - 18 0 ^{\circ}}{18} )}{sin ( θ ) cos ( \frac{36 θ - 18 0 ^{\circ}}{18} )} + \frac{sin ( \frac{- 18 0 ^{\circ} + 36 θ}{18} ) sin ( θ )}{cos ( \frac{- 18 0 ^{\circ} + 36 θ}{18} ) sin ( θ )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{- cos ( θ ) cos ( \frac{36 θ - 18 0 ^{\circ}}{18} ) + sin ( \frac{- 18 0 ^{\circ} + 36 θ}{18} ) sin ( θ )}{sin ( θ ) cos ( \frac{36 θ - 18 0 ^{\circ}}{18} )}

= \frac{- cos ( θ ) cos ( \frac{36 θ - 18 0 ^{\circ}}{18} ) + sin ( \frac{- 18 0 ^{\circ} + 36 θ}{18} ) sin ( θ )}{sin ( θ ) cos ( \frac{36 θ - 18 0 ^{\circ}}{18} )}

\frac{- cos ( \frac{- 18 0 ^{\circ} + 36 θ}{18} ) cos ( θ ) + sin ( \frac{- 18 0 ^{\circ} + 36 θ}{18} ) sin ( θ )}{cos ( \frac{- 18 0 ^{\circ} + 36 θ}{18} ) sin ( θ )} = 0

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

- cos (\frac{- 18 0 ^{\circ} + 36 θ}{18}) cos (θ) + sin (\frac{- 18 0 ^{\circ} + 36 θ}{18}) sin (θ) = 0

Rewrite using trig identities

- cos (\frac{- 18 0 ^{\circ} + 36 θ}{18}) cos (θ) + sin (\frac{- 18 0 ^{\circ} + 36 θ}{18}) sin (θ)

cos (s) cos (t) - sin (s) sin (t) = cos (s + t)

:فعّل متطابقة الجمع لزوايا

- cos (s) cos (t) + sin (s) sin (t) = - cos (s + t)

= - cos (\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ)

- cos (\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ) = 0

- 1

اقسم الطرفين على

- cos (\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ) = 0

- 1

اقسم الطرفين على

\frac{- cos ( \frac{- 18 0 ^{\circ} + 36 θ}{18} + θ )}{- 1} = \frac{0}{- 1}

بسّط

cos (\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ) = 0

cos (\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ) = 0

cos (\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ) = 0 :

حلول عامّة لـ

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ = 9 0^{\circ} + 36 0^{\circ} n, \frac{- 18 0 ^{\circ} + 36 θ}{18} + θ = 27 0^{\circ} + 36 0^{\circ} n

\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ = 9 0^{\circ} + 36 0^{\circ} n, \frac{- 18 0 ^{\circ} + 36 θ}{18} + θ = 27 0^{\circ} + 36 0^{\circ} n

\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ = 9 0^{\circ} + 36 0^{\circ} n

حلّ:

θ = 33.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ = 9 0^{\circ} + 36 0^{\circ} n

18

اضرب الطرفين بـ

\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ = 9 0^{\circ} + 36 0^{\circ} n

18

اضرب الطرفين بـ

\frac{- 18 0 ^{\circ} + 36 θ}{18} \cdot 18 + θ \cdot 18 = 9 0^{\circ} \cdot 18 + 36 0^{\circ} n \cdot 18

بسّط

\frac{- 18 0 ^{\circ} + 36 θ}{18} \cdot 18 + θ \cdot 18 = 9 0^{\circ} \cdot 18 + 36 0^{\circ} n \cdot 18

\frac{- 18 0 ^{\circ} + 36 θ}{18} \cdot 18

بسّط:

- 18 0^{\circ} + 36 θ

\frac{- 18 0 ^{\circ} + 36 θ}{18} \cdot 18

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

:اضرب كسور

= \frac{( - 18 0 ^{\circ} + 36 θ ) \cdot 18}{18}

18 :

إلغ العوامل المشتركة

= - - 18 0^{\circ} + 36 θ

θ \cdot 18

بسّط:

18 θ

θ \cdot 18

Apply the commutative law:

θ \cdot 18 = 18 θ

18 θ

9 0^{\circ} \cdot 18

بسّط:

162 0^{\circ}

9 0^{\circ} \cdot 18

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

:اضرب كسور

= 162 0^{\circ}

\frac{18}{2} = 9 :

اقسم الأعداد

= 162 0^{\circ}

36 0^{\circ} n \cdot 18

بسّط:

648 0^{\circ} n

36 0^{\circ} n \cdot 18

2 \cdot 18 = 36 :

اضرب الأعداد

= 648 0^{\circ} n

- 18 0^{\circ} + 36 θ + 18 θ = 162 0^{\circ} + 648 0^{\circ} n

- 18 0^{\circ} + 54 θ = 162 0^{\circ} + 648 0^{\circ} n

- 18 0^{\circ} + 54 θ = 162 0^{\circ} + 648 0^{\circ} n

- 18 0^{\circ} + 54 θ = 162 0^{\circ} + 648 0^{\circ} n

انقل

18 0^{\circ}

إلى الجانب الأيمن

- 18 0^{\circ} + 54 θ = 162 0^{\circ} + 648 0^{\circ} n

للطرفين

18 0^{\circ}

أضف

- 18 0^{\circ} + 54 θ + 18 0^{\circ} = 162 0^{\circ} + 648 0^{\circ} n + 18 0^{\circ}

بسّط

54 θ = 180 0^{\circ} + 648 0^{\circ} n

54 θ = 180 0^{\circ} + 648 0^{\circ} n

54

اقسم الطرفين على

54 θ = 180 0^{\circ} + 648 0^{\circ} n

54

اقسم الطرفين على

\frac{54 θ}{54} = 33.33333 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54}

بسّط

\frac{54 θ}{54} = 33.33333 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54}

\frac{54 θ}{54}

بسّط:

θ

\frac{54 θ}{54}

\frac{54}{54} = 1 :

اقسم الأعداد

= θ

33.33333 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54}

بسّط:

33.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

33.33333 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54}

33.33333 \dots^{\circ}

اختزل:

33.33333 \dots^{\circ}

33.33333 \dots^{\circ}

2 :

إلغ العوامل المشتركة

= 33.33333 \dots^{\circ}

= 33.33333 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54}

\frac{648 0 ^{\circ} n}{54}

اختزل:

\frac{36 0 ^{\circ} n}{3}

\frac{648 0 ^{\circ} n}{54}

18 :

إلغ العوامل المشتركة

= \frac{36 0 ^{\circ} n}{3}

= 33.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

θ = 33.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

θ = 33.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

θ = 33.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ = 27 0^{\circ} + 36 0^{\circ} n

حلّ:

θ = 93.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ = 27 0^{\circ} + 36 0^{\circ} n

18

اضرب الطرفين بـ

\frac{- 18 0 ^{\circ} + 36 θ}{18} + θ = 27 0^{\circ} + 36 0^{\circ} n

18

اضرب الطرفين بـ

\frac{- 18 0 ^{\circ} + 36 θ}{18} \cdot 18 + θ \cdot 18 = 27 0^{\circ} \cdot 18 + 36 0^{\circ} n \cdot 18

بسّط

\frac{- 18 0 ^{\circ} + 36 θ}{18} \cdot 18 + θ \cdot 18 = 27 0^{\circ} \cdot 18 + 36 0^{\circ} n \cdot 18

\frac{- 18 0 ^{\circ} + 36 θ}{18} \cdot 18

بسّط:

- 18 0^{\circ} + 36 θ

\frac{- 18 0 ^{\circ} + 36 θ}{18} \cdot 18

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

:اضرب كسور

= \frac{( - 18 0 ^{\circ} + 36 θ ) \cdot 18}{18}

18 :

إلغ العوامل المشتركة

= - - 18 0^{\circ} + 36 θ

θ \cdot 18

بسّط:

18 θ

θ \cdot 18

Apply the commutative law:

θ \cdot 18 = 18 θ

18 θ

27 0^{\circ} \cdot 18

بسّط:

486 0^{\circ}

27 0^{\circ} \cdot 18

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

:اضرب كسور

= 486 0^{\circ}

3 \cdot 18 = 54 :

اضرب الأعداد

= 486 0^{\circ}

\frac{54}{2} = 27 :

اقسم الأعداد

= 486 0^{\circ}

36 0^{\circ} n \cdot 18

بسّط:

648 0^{\circ} n

36 0^{\circ} n \cdot 18

2 \cdot 18 = 36 :

اضرب الأعداد

= 648 0^{\circ} n

- 18 0^{\circ} + 36 θ + 18 θ = 486 0^{\circ} + 648 0^{\circ} n

- 18 0^{\circ} + 54 θ = 486 0^{\circ} + 648 0^{\circ} n

- 18 0^{\circ} + 54 θ = 486 0^{\circ} + 648 0^{\circ} n

- 18 0^{\circ} + 54 θ = 486 0^{\circ} + 648 0^{\circ} n

انقل

18 0^{\circ}

إلى الجانب الأيمن

- 18 0^{\circ} + 54 θ = 486 0^{\circ} + 648 0^{\circ} n

للطرفين

18 0^{\circ}

أضف

- 18 0^{\circ} + 54 θ + 18 0^{\circ} = 486 0^{\circ} + 648 0^{\circ} n + 18 0^{\circ}

بسّط

54 θ = 504 0^{\circ} + 648 0^{\circ} n

54 θ = 504 0^{\circ} + 648 0^{\circ} n

54

اقسم الطرفين على

54 θ = 504 0^{\circ} + 648 0^{\circ} n

54

اقسم الطرفين على

\frac{54 θ}{54} = 93.33333 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54}

بسّط

\frac{54 θ}{54} = 93.33333 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54}

\frac{54 θ}{54}

بسّط:

θ

\frac{54 θ}{54}

\frac{54}{54} = 1 :

اقسم الأعداد

= θ

93.33333 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54}

بسّط:

93.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

93.33333 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54}

93.33333 \dots^{\circ}

اختزل:

93.33333 \dots^{\circ}

93.33333 \dots^{\circ}

2 :

إلغ العوامل المشتركة

= 93.33333 \dots^{\circ}

= 93.33333 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54}

\frac{648 0 ^{\circ} n}{54}

اختزل:

\frac{36 0 ^{\circ} n}{3}

\frac{648 0 ^{\circ} n}{54}

18 :

إلغ العوامل المشتركة

= \frac{36 0 ^{\circ} n}{3}

= 93.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

θ = 93.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

θ = 93.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

θ = 93.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

θ = 33.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}, θ = 93.33333 \dots^{\circ} + \frac{36 0 ^{\circ} n}{3}

رسم

أمثلة شائعة

sinh(x)=4

sinh (x) = 4

2cos^2(θ)-3cos(θ)+1=0,0<= θ<2pi

2 cos^{2} (θ) - 3 cos (θ) + 1 = 0, 0 \leq θ < 2 π

cos(x)sin(x)=1

cos (x) sin (x) = 1

24arctan(x)=4pi

24 arctan (x) = 4 π

cos^2(x)-sin(x)-1=0

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