English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

شائع علم المثلثات >

tan(3x)*cot(x+40)=1

الحلّ

tan (3 x) \cdot cot (x + 4 0^{\circ}) = 1

الحلّ

x = 18 0^{\circ} n + 2 0^{\circ}, x = 11 0^{\circ} + 18 0^{\circ} n

راديان

x = \frac{π}{9} + πn, x = \frac{11 π}{18} + πn

خطوات الحلّ

tan (3 x) cot (x + 4 0^{\circ}) = 1

من الطرفين

1

اطرح

tan (3 x) cot (x + 4 0^{\circ}) - 1 = 0

sin, cos :

عبّر بواسطة

- 1 + cot (4 0^{\circ} + x) tan (3 x)

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

:Use the basic trigonometric identity

= - 1 + \frac{cos ( 4 0 ^{\circ} + x )}{sin ( 4 0 ^{\circ} + x )} tan (3 x)

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

:Use the basic trigonometric identity

= - 1 + \frac{cos ( 4 0 ^{\circ} + x )}{sin ( 4 0 ^{\circ} + x )} \cdot \frac{sin ( 3 x )}{cos ( 3 x )}

- 1 + \frac{cos ( 4 0 ^{\circ} + x )}{sin ( 4 0 ^{\circ} + x )} \cdot \frac{sin ( 3 x )}{cos ( 3 x )}

بسّط:

\frac{- sin ( \frac{36 0 ^{\circ} + 9 x}{9} ) cos ( 3 x ) + cos ( \frac{36 0 ^{\circ} + 9 x}{9} ) sin ( 3 x )}{sin ( \frac{36 0 ^{\circ} + 9 x}{9} ) cos ( 3 x )}

- 1 + \frac{cos ( 4 0 ^{\circ} + x )}{sin ( 4 0 ^{\circ} + x )} \cdot \frac{sin ( 3 x )}{cos ( 3 x )}

\frac{cos ( 4 0 ^{\circ} + x )}{sin ( 4 0 ^{\circ} + x )} \cdot \frac{sin ( 3 x )}{cos ( 3 x )} = \frac{cos ( \frac{36 0 ^{\circ} + 9 x}{9} ) sin ( 3 x )}{sin ( \frac{36 0 ^{\circ} + 9 x}{9} ) cos ( 3 x )}

\frac{cos ( 4 0 ^{\circ} + x )}{sin ( 4 0 ^{\circ} + x )} \cdot \frac{sin ( 3 x )}{cos ( 3 x )}

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

:اضرب كسور

= \frac{cos ( 4 0 ^{\circ} + x ) sin ( 3 x )}{sin ( 4 0 ^{\circ} + x ) cos ( 3 x )}

4 0^{\circ} + x

وحّد:

\frac{36 0 ^{\circ} + 9 x}{9}

4 0^{\circ} + x

x = \frac{x 9}{9}

:حوّل الأعداد لكسور

= 4 0^{\circ} + \frac{x \cdot 9}{9}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{36 0 ^{\circ} + x \cdot 9}{9}

= \frac{cos ( x + 4 0 ^{\circ} ) sin ( 3 x )}{sin ( \frac{9 x + 36 0 ^{\circ}}{9} ) cos ( 3 x )}

4 0^{\circ} + x

وحّد:

\frac{36 0 ^{\circ} + 9 x}{9}

4 0^{\circ} + x

x = \frac{x 9}{9}

:حوّل الأعداد لكسور

= 4 0^{\circ} + \frac{x \cdot 9}{9}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{36 0 ^{\circ} + x \cdot 9}{9}

= \frac{cos ( \frac{9 x + 36 0 ^{\circ}}{9} ) sin ( 3 x )}{sin ( \frac{9 x + 36 0 ^{\circ}}{9} ) cos ( 3 x )}

= - 1 + \frac{cos ( \frac{9 x + 36 0 ^{\circ}}{9} ) sin ( 3 x )}{sin ( \frac{9 x + 36 0 ^{\circ}}{9} ) cos ( 3 x )}

1 = \frac{1 sin ( \frac{36 0 ^{\circ} + x 9}{9} ) cos ( 3 x )}{sin ( \frac{36 0 ^{\circ} + x 9}{9} ) cos ( 3 x )}

:حوّل الأعداد لكسور

= - \frac{1 \cdot sin ( \frac{36 0 ^{\circ} + x \cdot 9}{9} ) cos ( 3 x )}{sin ( \frac{36 0 ^{\circ} + x \cdot 9}{9} ) cos ( 3 x )} + \frac{cos ( \frac{36 0 ^{\circ} + x \cdot 9}{9} ) sin ( 3 x )}{sin ( \frac{36 0 ^{\circ} + x \cdot 9}{9} ) cos ( 3 x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{- 1 \cdot sin ( \frac{36 0 ^{\circ} + x \cdot 9}{9} ) cos ( 3 x ) + cos ( \frac{36 0 ^{\circ} + x \cdot 9}{9} ) sin ( 3 x )}{sin ( \frac{36 0 ^{\circ} + x \cdot 9}{9} ) cos ( 3 x )}

1 \cdot sin (\frac{36 0 ^{\circ} + x \cdot 9}{9}) = sin (\frac{36 0 ^{\circ} + x \cdot 9}{9}) :

اضرب

= \frac{- sin ( \frac{9 x + 36 0 ^{\circ}}{9} ) cos ( 3 x ) + cos ( \frac{9 x + 36 0 ^{\circ}}{9} ) sin ( 3 x )}{sin ( \frac{9 x + 36 0 ^{\circ}}{9} ) cos ( 3 x )}

= \frac{- sin ( \frac{36 0 ^{\circ} + 9 x}{9} ) cos ( 3 x ) + cos ( \frac{36 0 ^{\circ} + 9 x}{9} ) sin ( 3 x )}{sin ( \frac{36 0 ^{\circ} + 9 x}{9} ) cos ( 3 x )}

\frac{- cos ( 3 x ) sin ( \frac{36 0 ^{\circ} + 9 x}{9} ) + cos ( \frac{36 0 ^{\circ} + 9 x}{9} ) sin ( 3 x )}{cos ( 3 x ) sin ( \frac{36 0 ^{\circ} + 9 x}{9} )} = 0

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

- cos (3 x) sin (\frac{36 0 ^{\circ} + 9 x}{9}) + cos (\frac{36 0 ^{\circ} + 9 x}{9}) sin (3 x) = 0

Rewrite using trig identities

- cos (3 x) sin (\frac{36 0 ^{\circ} + 9 x}{9}) + cos (\frac{36 0 ^{\circ} + 9 x}{9}) sin (3 x)

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

:فعّل متطابقة الطرح لزوايا

= sin (3 x - \frac{36 0 ^{\circ} + 9 x}{9})

sin (3 x - \frac{36 0 ^{\circ} + 9 x}{9}) = 0

sin (3 x - \frac{36 0 ^{\circ} + 9 x}{9}) = 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}

3 x - \frac{36 0 ^{\circ} + 9 x}{9} = 0 + 36 0^{\circ} n, 3 x - \frac{36 0 ^{\circ} + 9 x}{9} = 18 0^{\circ} + 36 0^{\circ} n

3 x - \frac{36 0 ^{\circ} + 9 x}{9} = 0 + 36 0^{\circ} n, 3 x - \frac{36 0 ^{\circ} + 9 x}{9} = 18 0^{\circ} + 36 0^{\circ} n

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

حلّ:

x = 18 0^{\circ} n + 2 0^{\circ}

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

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

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

9

اضرب الطرفين بـ

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

9

اضرب الطرفين بـ

3 x \cdot 9 - \frac{36 0 ^{\circ} + 9 x}{9} \cdot 9 = 36 0^{\circ} n \cdot 9

بسّط

3 x \cdot 9 - \frac{36 0 ^{\circ} + 9 x}{9} \cdot 9 = 36 0^{\circ} n \cdot 9

3 x \cdot 9

بسّط:

27 x

3 x \cdot 9

3 \cdot 9 = 27 :

اضرب الأعداد

= 27 x

- \frac{36 0 ^{\circ} + 9 x}{9} \cdot 9

بسّط:

- (36 0^{\circ} + 9 x)

- \frac{36 0 ^{\circ} + 9 x}{9} \cdot 9

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

:اضرب كسور

= - \frac{( 36 0 ^{\circ} + 9 x ) \cdot 9}{9}

9 :

إلغ العوامل المشتركة

= - (9 x + 36 0^{\circ})

36 0^{\circ} n \cdot 9

بسّط:

324 0^{\circ} n

36 0^{\circ} n \cdot 9

2 \cdot 9 = 18 :

اضرب الأعداد

= 324 0^{\circ} n

27 x - (36 0^{\circ} + 9 x) = 324 0^{\circ} n

27 x - (36 0^{\circ} + 9 x) = 324 0^{\circ} n

27 x - (36 0^{\circ} + 9 x) = 324 0^{\circ} n

27 x - (36 0^{\circ} + 9 x)

وسّع:

18 x - 36 0^{\circ}

27 x - (36 0^{\circ} + 9 x)

- (36 0^{\circ} + 9 x) : - 36 0^{\circ} - 9 x

- (36 0^{\circ} + 9 x)

افتح أقواس

= - (36 0^{\circ}) - (9 x)

فعّل قوانين سالب-موجب

+ (- a) = - a

= - 36 0^{\circ} - 9 x

= 27 x - 36 0^{\circ} - 9 x

27 x - 36 0^{\circ} - 9 x

بسّط:

18 x - 36 0^{\circ}

27 x - 36 0^{\circ} - 9 x

جمّع التعابير المتشابهة

= 27 x - 9 x - 36 0^{\circ}

27 x - 9 x = 18 x :

اجمع العناصر المتشابهة

= 18 x - 36 0^{\circ}

= 18 x - 36 0^{\circ}

18 x - 36 0^{\circ} = 324 0^{\circ} n

انقل

36 0^{\circ}

إلى الجانب الأيمن

18 x - 36 0^{\circ} = 324 0^{\circ} n

للطرفين

36 0^{\circ}

أضف

18 x - 36 0^{\circ} + 36 0^{\circ} = 324 0^{\circ} n + 36 0^{\circ}

بسّط

18 x = 324 0^{\circ} n + 36 0^{\circ}

18 x = 324 0^{\circ} n + 36 0^{\circ}

18

اقسم الطرفين على

18 x = 324 0^{\circ} n + 36 0^{\circ}

18

اقسم الطرفين على

\frac{18 x}{18} = \frac{324 0 ^{\circ} n}{18} + 2 0^{\circ}

بسّط

\frac{18 x}{18} = \frac{324 0 ^{\circ} n}{18} + 2 0^{\circ}

\frac{18 x}{18}

بسّط:

x

\frac{18 x}{18}

\frac{18}{18} = 1 :

اقسم الأعداد

= x

\frac{324 0 ^{\circ} n}{18} + 2 0^{\circ}

بسّط:

18 0^{\circ} n + 2 0^{\circ}

\frac{324 0 ^{\circ} n}{18} + 2 0^{\circ}

\frac{18}{18} = 1 :

اقسم الأعداد

= 18 0^{\circ} n + 2 0^{\circ}

2 0^{\circ}

اختزل:

2 0^{\circ}

2 0^{\circ}

2 :

إلغ العوامل المشتركة

= 2 0^{\circ}

= 18 0^{\circ} n + 2 0^{\circ}

x = 18 0^{\circ} n + 2 0^{\circ}

x = 18 0^{\circ} n + 2 0^{\circ}

x = 18 0^{\circ} n + 2 0^{\circ}

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

حلّ:

x = 11 0^{\circ} + 18 0^{\circ} n

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

9

اضرب الطرفين بـ

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

9

اضرب الطرفين بـ

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

بسّط

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

3 x \cdot 9

بسّط:

27 x

3 x \cdot 9

3 \cdot 9 = 27 :

اضرب الأعداد

= 27 x

- \frac{36 0 ^{\circ} + 9 x}{9} \cdot 9

بسّط:

- (36 0^{\circ} + 9 x)

- \frac{36 0 ^{\circ} + 9 x}{9} \cdot 9

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

:اضرب كسور

= - \frac{( 36 0 ^{\circ} + 9 x ) \cdot 9}{9}

9 :

إلغ العوامل المشتركة

= - (9 x + 36 0^{\circ})

18 0^{\circ} 9

بسّط:

162 0^{\circ}

18 0^{\circ} 9

Apply the commutative law:

18 0^{\circ} 9 = 162 0^{\circ}

162 0^{\circ}

36 0^{\circ} n \cdot 9

بسّط:

324 0^{\circ} n

36 0^{\circ} n \cdot 9

2 \cdot 9 = 18 :

اضرب الأعداد

= 324 0^{\circ} n

27 x - (36 0^{\circ} + 9 x) = 162 0^{\circ} + 324 0^{\circ} n

27 x - (36 0^{\circ} + 9 x) = 162 0^{\circ} + 324 0^{\circ} n

27 x - (36 0^{\circ} + 9 x) = 162 0^{\circ} + 324 0^{\circ} n

27 x - (36 0^{\circ} + 9 x)

وسّع:

18 x - 36 0^{\circ}

27 x - (36 0^{\circ} + 9 x)

- (36 0^{\circ} + 9 x) : - 36 0^{\circ} - 9 x

- (36 0^{\circ} + 9 x)

افتح أقواس

= - (36 0^{\circ}) - (9 x)

فعّل قوانين سالب-موجب

+ (- a) = - a

= - 36 0^{\circ} - 9 x

= 27 x - 36 0^{\circ} - 9 x

27 x - 36 0^{\circ} - 9 x

بسّط:

18 x - 36 0^{\circ}

27 x - 36 0^{\circ} - 9 x

جمّع التعابير المتشابهة

= 27 x - 9 x - 36 0^{\circ}

27 x - 9 x = 18 x :

اجمع العناصر المتشابهة

= 18 x - 36 0^{\circ}

= 18 x - 36 0^{\circ}

18 x - 36 0^{\circ} = 162 0^{\circ} + 324 0^{\circ} n

انقل

36 0^{\circ}

إلى الجانب الأيمن

18 x - 36 0^{\circ} = 162 0^{\circ} + 324 0^{\circ} n

للطرفين

36 0^{\circ}

أضف

18 x - 36 0^{\circ} + 36 0^{\circ} = 162 0^{\circ} + 324 0^{\circ} n + 36 0^{\circ}

بسّط

18 x = 198 0^{\circ} + 324 0^{\circ} n

18 x = 198 0^{\circ} + 324 0^{\circ} n

18

اقسم الطرفين على

18 x = 198 0^{\circ} + 324 0^{\circ} n

18

اقسم الطرفين على

\frac{18 x}{18} = 11 0^{\circ} + \frac{324 0 ^{\circ} n}{18}

بسّط

x = 11 0^{\circ} + 18 0^{\circ} n

x = 11 0^{\circ} + 18 0^{\circ} n

x = 18 0^{\circ} n + 2 0^{\circ}, x = 11 0^{\circ} + 18 0^{\circ} n

رسم

أمثلة شائعة

cos(x)=-1/2 sqrt(2)

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

48sin^2(x)=48-24cos(x)

48 sin^{2} (x) = 48 - 24 cos (x)

1-4cos(2x)=0

1 - 4 cos (2 x) = 0

tan^2(x)+5tan(x)=0

tan^{2} (x) + 5 tan (x) = 0

sin^3(x)=cos^2(x)

sin^{3} (x) = cos^{2} (x)