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شائع علم المثلثات >

tan(4x+20)*cot(x+50)=1

الحلّ

tan (4 x + 20) \cdot cot (x + 5 0^{\circ}) = 1

الحلّ

x = \frac{36 0 ^{\circ} n}{3} - \frac{20}{3} + 16.66666 \dots^{\circ}, x = 76.66666 \dots^{\circ} - \frac{20}{3} + \frac{36 0 ^{\circ} n}{3}

راديان

x = - \frac{20}{3} + \frac{5 π}{54} + \frac{2 π}{3} n, x = \frac{23 π}{54} - \frac{20}{3} + \frac{2 π}{3} n

خطوات الحلّ

tan (4 x + 20) cot (x + 5 0^{\circ}) = 1

من الطرفين

1

اطرح

tan (4 x + 20) cot (x + 5 0^{\circ}) - 1 = 0

sin, cos :

عبّر بواسطة

- 1 + cot (5 0^{\circ} + x) tan (20 + 4 x)

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

:Use the basic trigonometric identity

= - 1 + \frac{cos ( 5 0 ^{\circ} + x )}{sin ( 5 0 ^{\circ} + x )} tan (20 + 4 x)

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

:Use the basic trigonometric identity

= - 1 + \frac{cos ( 5 0 ^{\circ} + x )}{sin ( 5 0 ^{\circ} + x )} \cdot \frac{sin ( 20 + 4 x )}{cos ( 20 + 4 x )}

- 1 + \frac{cos ( 5 0 ^{\circ} + x )}{sin ( 5 0 ^{\circ} + x )} \cdot \frac{sin ( 20 + 4 x )}{cos ( 20 + 4 x )}

بسّط:

\frac{- sin ( \frac{90 0 ^{\circ} + 18 x}{18} ) cos ( 20 + 4 x ) + cos ( \frac{90 0 ^{\circ} + 18 x}{18} ) sin ( 20 + 4 x )}{sin ( \frac{90 0 ^{\circ} + 18 x}{18} ) cos ( 20 + 4 x )}

- 1 + \frac{cos ( 5 0 ^{\circ} + x )}{sin ( 5 0 ^{\circ} + x )} \cdot \frac{sin ( 20 + 4 x )}{cos ( 20 + 4 x )}

\frac{cos ( 5 0 ^{\circ} + x )}{sin ( 5 0 ^{\circ} + x )} \cdot \frac{sin ( 20 + 4 x )}{cos ( 20 + 4 x )} = \frac{cos ( \frac{90 0 ^{\circ} + 18 x}{18} ) sin ( 20 + 4 x )}{sin ( \frac{90 0 ^{\circ} + 18 x}{18} ) cos ( 20 + 4 x )}

\frac{cos ( 5 0 ^{\circ} + x )}{sin ( 5 0 ^{\circ} + x )} \cdot \frac{sin ( 20 + 4 x )}{cos ( 20 + 4 x )}

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

:اضرب كسور

= \frac{cos ( 5 0 ^{\circ} + x ) sin ( 20 + 4 x )}{sin ( 5 0 ^{\circ} + x ) cos ( 20 + 4 x )}

5 0^{\circ} + x

وحّد:

\frac{90 0 ^{\circ} + 18 x}{18}

5 0^{\circ} + x

x = \frac{x 18}{18}

:حوّل الأعداد لكسور

= 5 0^{\circ} + \frac{x \cdot 18}{18}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{90 0 ^{\circ} + x \cdot 18}{18}

= \frac{cos ( x + 5 0 ^{\circ} ) sin ( 4 x + 20 )}{sin ( \frac{18 x + 90 0 ^{\circ}}{18} ) cos ( 4 x + 20 )}

5 0^{\circ} + x

وحّد:

\frac{90 0 ^{\circ} + 18 x}{18}

5 0^{\circ} + x

x = \frac{x 18}{18}

:حوّل الأعداد لكسور

= 5 0^{\circ} + \frac{x \cdot 18}{18}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{90 0 ^{\circ} + x \cdot 18}{18}

= \frac{cos ( \frac{18 x + 90 0 ^{\circ}}{18} ) sin ( 4 x + 20 )}{sin ( \frac{18 x + 90 0 ^{\circ}}{18} ) cos ( 4 x + 20 )}

= - 1 + \frac{cos ( \frac{18 x + 90 0 ^{\circ}}{18} ) sin ( 4 x + 20 )}{sin ( \frac{18 x + 90 0 ^{\circ}}{18} ) cos ( 4 x + 20 )}

1 = \frac{1 sin ( \frac{90 0 ^{\circ} + x 18}{18} ) cos ( 20 + 4 x )}{sin ( \frac{90 0 ^{\circ} + x 18}{18} ) cos ( 20 + 4 x )}

:حوّل الأعداد لكسور

= - \frac{1 \cdot sin ( \frac{90 0 ^{\circ} + x \cdot 18}{18} ) cos ( 20 + 4 x )}{sin ( \frac{90 0 ^{\circ} + x \cdot 18}{18} ) cos ( 20 + 4 x )} + \frac{cos ( \frac{90 0 ^{\circ} + x \cdot 18}{18} ) sin ( 20 + 4 x )}{sin ( \frac{90 0 ^{\circ} + x \cdot 18}{18} ) cos ( 20 + 4 x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{- 1 \cdot sin ( \frac{90 0 ^{\circ} + x \cdot 18}{18} ) cos ( 20 + 4 x ) + cos ( \frac{90 0 ^{\circ} + x \cdot 18}{18} ) sin ( 20 + 4 x )}{sin ( \frac{90 0 ^{\circ} + x \cdot 18}{18} ) cos ( 20 + 4 x )}

1 \cdot sin (\frac{90 0 ^{\circ} + x \cdot 18}{18}) = sin (\frac{90 0 ^{\circ} + x \cdot 18}{18}) :

اضرب

= \frac{- sin ( \frac{18 x + 90 0 ^{\circ}}{18} ) cos ( 4 x + 20 ) + cos ( \frac{18 x + 90 0 ^{\circ}}{18} ) sin ( 4 x + 20 )}{sin ( \frac{18 x + 90 0 ^{\circ}}{18} ) cos ( 4 x + 20 )}

= \frac{- sin ( \frac{90 0 ^{\circ} + 18 x}{18} ) cos ( 20 + 4 x ) + cos ( \frac{90 0 ^{\circ} + 18 x}{18} ) sin ( 20 + 4 x )}{sin ( \frac{90 0 ^{\circ} + 18 x}{18} ) cos ( 20 + 4 x )}

\frac{- cos ( 20 + 4 x ) sin ( \frac{18 x + 90 0 ^{\circ}}{18} ) + cos ( \frac{18 x + 90 0 ^{\circ}}{18} ) sin ( 20 + 4 x )}{cos ( 20 + 4 x ) sin ( \frac{18 x + 90 0 ^{\circ}}{18} )} = 0

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

- cos (20 + 4 x) sin (\frac{18 x + 90 0 ^{\circ}}{18}) + cos (\frac{18 x + 90 0 ^{\circ}}{18}) sin (20 + 4 x) = 0

Rewrite using trig identities

- cos (20 + 4 x) sin (\frac{18 x + 90 0 ^{\circ}}{18}) + cos (\frac{18 x + 90 0 ^{\circ}}{18}) sin (20 + 4 x)

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

:فعّل متطابقة الطرح لزوايا

= sin (20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18})

sin (20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18}) = 0

sin (20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18}) = 0 :

حلول عامّة لـ

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 0 + 36 0^{\circ} n, 20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 18 0^{\circ} + 36 0^{\circ} n

20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 0 + 36 0^{\circ} n, 20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 18 0^{\circ} + 36 0^{\circ} n

20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 0 + 36 0^{\circ} n

حلّ:

x = \frac{36 0 ^{\circ} n}{3} - \frac{20}{3} + 16.66666 \dots^{\circ}

20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 0 + 36 0^{\circ} n

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

20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 36 0^{\circ} n

انقل

20

إلى الجانب الأيمن

20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 36 0^{\circ} n

من الطرفين

20

اطرح

20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18} - 20 = 36 0^{\circ} n - 20

بسّط

4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 36 0^{\circ} n - 20

4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 36 0^{\circ} n - 20

18

اضرب الطرفين بـ

4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 36 0^{\circ} n - 20

18

اضرب الطرفين بـ

4 x \cdot 18 - \frac{18 x + 90 0 ^{\circ}}{18} \cdot 18 = 36 0^{\circ} n \cdot 18 - 20 \cdot 18

بسّط

4 x \cdot 18 - \frac{18 x + 90 0 ^{\circ}}{18} \cdot 18 = 36 0^{\circ} n \cdot 18 - 20 \cdot 18

4 x \cdot 18

بسّط:

72 x

4 x \cdot 18

4 \cdot 18 = 72 :

اضرب الأعداد

= 72 x

- \frac{18 x + 90 0 ^{\circ}}{18} \cdot 18

بسّط:

- (18 x + 90 0^{\circ})

- \frac{18 x + 90 0 ^{\circ}}{18} \cdot 18

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

:اضرب كسور

= - \frac{( 18 x + 90 0 ^{\circ} ) \cdot 18}{18}

18 :

إلغ العوامل المشتركة

= - (18 x + 90 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

- 20 \cdot 18

بسّط:

- 360

- 20 \cdot 18

20 \cdot 18 = 360 :

اضرب الأعداد

= - 360

72 x - (18 x + 90 0^{\circ}) = 648 0^{\circ} n - 360

72 x - (18 x + 90 0^{\circ}) = 648 0^{\circ} n - 360

72 x - (18 x + 90 0^{\circ}) = 648 0^{\circ} n - 360

72 x - (18 x + 90 0^{\circ})

وسّع:

54 x - 90 0^{\circ}

72 x - (18 x + 90 0^{\circ})

- (18 x + 90 0^{\circ}) : - 18 x - 90 0^{\circ}

- (18 x + 90 0^{\circ})

افتح أقواس

= - (18 x) - (90 0^{\circ})

فعّل قوانين سالب-موجب

+ (- a) = - a

= - 18 x - 90 0^{\circ}

= 72 x - 18 x - 90 0^{\circ}

72 x - 18 x = 54 x :

اجمع العناصر المتشابهة

= 54 x - 90 0^{\circ}

54 x - 90 0^{\circ} = 648 0^{\circ} n - 360

انقل

90 0^{\circ}

إلى الجانب الأيمن

54 x - 90 0^{\circ} = 648 0^{\circ} n - 360

للطرفين

90 0^{\circ}

أضف

54 x - 90 0^{\circ} + 90 0^{\circ} = 648 0^{\circ} n - 360 + 90 0^{\circ}

بسّط

54 x = 648 0^{\circ} n - 360 + 90 0^{\circ}

54 x = 648 0^{\circ} n - 360 + 90 0^{\circ}

54

اقسم الطرفين على

54 x = 648 0^{\circ} n - 360 + 90 0^{\circ}

54

اقسم الطرفين على

\frac{54 x}{54} = \frac{648 0 ^{\circ} n}{54} - \frac{360}{54} + 16.66666 \dots^{\circ}

بسّط

\frac{54 x}{54} = \frac{648 0 ^{\circ} n}{54} - \frac{360}{54} + 16.66666 \dots^{\circ}

\frac{54 x}{54}

بسّط:

x

\frac{54 x}{54}

\frac{54}{54} = 1 :

اقسم الأعداد

= x

\frac{648 0 ^{\circ} n}{54} - \frac{360}{54} + 16.66666 \dots^{\circ}

بسّط:

\frac{36 0 ^{\circ} n}{3} - \frac{20}{3} + 16.66666 \dots^{\circ}

\frac{648 0 ^{\circ} n}{54} - \frac{360}{54} + 16.66666 \dots^{\circ}

\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}

= \frac{36 0 ^{\circ} n}{3} - \frac{360}{54} + 16.66666 \dots^{\circ}

\frac{360}{54}

اختزل:

\frac{20}{3}

\frac{360}{54}

18 :

إلغ العوامل المشتركة

= \frac{20}{3}

= \frac{36 0 ^{\circ} n}{3} - \frac{20}{3} + 16.66666 \dots^{\circ}

x = \frac{36 0 ^{\circ} n}{3} - \frac{20}{3} + 16.66666 \dots^{\circ}

x = \frac{36 0 ^{\circ} n}{3} - \frac{20}{3} + 16.66666 \dots^{\circ}

x = \frac{36 0 ^{\circ} n}{3} - \frac{20}{3} + 16.66666 \dots^{\circ}

20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 18 0^{\circ} + 36 0^{\circ} n

حلّ:

x = 76.66666 \dots^{\circ} - \frac{20}{3} + \frac{36 0 ^{\circ} n}{3}

20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 18 0^{\circ} + 36 0^{\circ} n

انقل

20

إلى الجانب الأيمن

20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 18 0^{\circ} + 36 0^{\circ} n

من الطرفين

20

اطرح

20 + 4 x - \frac{18 x + 90 0 ^{\circ}}{18} - 20 = 18 0^{\circ} + 36 0^{\circ} n - 20

بسّط

4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 18 0^{\circ} + 36 0^{\circ} n - 20

4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 18 0^{\circ} + 36 0^{\circ} n - 20

18

اضرب الطرفين بـ

4 x - \frac{18 x + 90 0 ^{\circ}}{18} = 18 0^{\circ} + 36 0^{\circ} n - 20

18

اضرب الطرفين بـ

4 x \cdot 18 - \frac{18 x + 90 0 ^{\circ}}{18} \cdot 18 = 18 0^{\circ} 18 + 36 0^{\circ} n \cdot 18 - 20 \cdot 18

بسّط

4 x \cdot 18 - \frac{18 x + 90 0 ^{\circ}}{18} \cdot 18 = 18 0^{\circ} 18 + 36 0^{\circ} n \cdot 18 - 20 \cdot 18

4 x \cdot 18

بسّط:

72 x

4 x \cdot 18

4 \cdot 18 = 72 :

اضرب الأعداد

= 72 x

- \frac{18 x + 90 0 ^{\circ}}{18} \cdot 18

بسّط:

- (18 x + 90 0^{\circ})

- \frac{18 x + 90 0 ^{\circ}}{18} \cdot 18

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

:اضرب كسور

= - \frac{( 18 x + 90 0 ^{\circ} ) \cdot 18}{18}

18 :

إلغ العوامل المشتركة

= - (18 x + 90 0^{\circ})

18 0^{\circ} 18

بسّط:

324 0^{\circ}

18 0^{\circ} 18

Apply the commutative law:

18 0^{\circ} 18 = 324 0^{\circ}

324 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

- 20 \cdot 18

بسّط:

- 360

- 20 \cdot 18

20 \cdot 18 = 360 :

اضرب الأعداد

= - 360

72 x - (18 x + 90 0^{\circ}) = 324 0^{\circ} + 648 0^{\circ} n - 360

72 x - (18 x + 90 0^{\circ}) = 324 0^{\circ} + 648 0^{\circ} n - 360

72 x - (18 x + 90 0^{\circ}) = 324 0^{\circ} + 648 0^{\circ} n - 360

72 x - (18 x + 90 0^{\circ})

وسّع:

54 x - 90 0^{\circ}

72 x - (18 x + 90 0^{\circ})

- (18 x + 90 0^{\circ}) : - 18 x - 90 0^{\circ}

- (18 x + 90 0^{\circ})

افتح أقواس

= - (18 x) - (90 0^{\circ})

فعّل قوانين سالب-موجب

+ (- a) = - a

= - 18 x - 90 0^{\circ}

= 72 x - 18 x - 90 0^{\circ}

72 x - 18 x = 54 x :

اجمع العناصر المتشابهة

= 54 x - 90 0^{\circ}

54 x - 90 0^{\circ} = 324 0^{\circ} + 648 0^{\circ} n - 360

انقل

90 0^{\circ}

إلى الجانب الأيمن

54 x - 90 0^{\circ} = 324 0^{\circ} + 648 0^{\circ} n - 360

للطرفين

90 0^{\circ}

أضف

54 x - 90 0^{\circ} + 90 0^{\circ} = 324 0^{\circ} + 648 0^{\circ} n - 360 + 90 0^{\circ}

بسّط

54 x = 414 0^{\circ} + 648 0^{\circ} n - 360

54 x = 414 0^{\circ} + 648 0^{\circ} n - 360

54

اقسم الطرفين على

54 x = 414 0^{\circ} + 648 0^{\circ} n - 360

54

اقسم الطرفين على

\frac{54 x}{54} = 76.66666 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54} - \frac{360}{54}

بسّط

\frac{54 x}{54} = 76.66666 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54} - \frac{360}{54}

\frac{54 x}{54}

بسّط:

x

\frac{54 x}{54}

\frac{54}{54} = 1 :

اقسم الأعداد

= x

76.66666 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54} - \frac{360}{54}

بسّط:

76.66666 \dots^{\circ} - \frac{20}{3} + \frac{36 0 ^{\circ} n}{3}

76.66666 \dots^{\circ} + \frac{648 0 ^{\circ} n}{54} - \frac{360}{54}

جمّع التعابير المتشابهة

= 76.66666 \dots^{\circ} - \frac{360}{54} + \frac{648 0 ^{\circ} n}{54}

\frac{360}{54}

اختزل:

\frac{20}{3}

\frac{360}{54}

18 :

إلغ العوامل المشتركة

= \frac{20}{3}

= 76.66666 \dots^{\circ} - \frac{20}{3} + \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}

= 76.66666 \dots^{\circ} - \frac{20}{3} + \frac{36 0 ^{\circ} n}{3}

x = 76.66666 \dots^{\circ} - \frac{20}{3} + \frac{36 0 ^{\circ} n}{3}

x = 76.66666 \dots^{\circ} - \frac{20}{3} + \frac{36 0 ^{\circ} n}{3}

x = 76.66666 \dots^{\circ} - \frac{20}{3} + \frac{36 0 ^{\circ} n}{3}

x = \frac{36 0 ^{\circ} n}{3} - \frac{20}{3} + 16.66666 \dots^{\circ}, x = 76.66666 \dots^{\circ} - \frac{20}{3} + \frac{36 0 ^{\circ} n}{3}

رسم

أمثلة شائعة

csc(x)=-0.5

csc (x) = - 0.5

cos(θ)-4=-3

cos (θ) - 4 = - 3

csc^2(x)+2csc(x)-3=-4

csc^{2} (x) + 2 csc (x) - 3 = - 4

-cos(θ)=0

- cos (θ) = 0

cos(w)=0.53

cos (w) = 0.53