English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

شائع علم المثلثات >

3tan(x+43)=2cos(x+43)

الحلّ

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

الحلّ

x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

راديان

x = - \frac{13 π}{180} + 2 πn, x = \frac{107 π}{180} + 2 πn

خطوات الحلّ

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

من الطرفين

2 cos (x + 4 3^{\circ})

اطرح

3 tan (x + 4 3^{\circ}) - 2 cos (x + 4 3^{\circ}) = 0

3 tan (x + 4 3^{\circ}) - 2 cos (x + 4 3^{\circ})

بسّط:

3 tan (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180})

3 tan (x + 4 3^{\circ}) - 2 cos (x + 4 3^{\circ})

x + 4 3^{\circ}

وحّد:

\frac{180 x + 774 0 ^{\circ}}{180}

x + 4 3^{\circ}

x = \frac{x 180}{180}

:حوّل الأعداد لكسور

= \frac{x \cdot 180}{180} + 4 3^{\circ}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{x \cdot 180 + 774 0 ^{\circ}}{180}

= 3 tan (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos (x + 4 3^{\circ})

x + 4 3^{\circ}

وحّد:

\frac{180 x + 774 0 ^{\circ}}{180}

x + 4 3^{\circ}

x = \frac{x 180}{180}

:حوّل الأعداد لكسور

= \frac{x \cdot 180}{180} + 4 3^{\circ}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{x \cdot 180 + 774 0 ^{\circ}}{180}

= 3 tan (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180})

3 tan (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

sin, cos :

عبّر بواسطة

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180})

بسّط:

\frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) - 2 cos ^{2} ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180})

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

اضرب بـ:

\frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

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

:اضرب كسور

= \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) \cdot 3}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

= \frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180})

2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{2 cos ( \frac{180 x + 774 0 ^{\circ}}{180} ) cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

:حوّل الأعداد لكسور

= \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) \cdot 3}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - \frac{2 cos ( \frac{180 x + 774 0 ^{\circ}}{180} ) cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) \cdot 3 - 2 cos ( \frac{180 x + 774 0 ^{\circ}}{180} ) cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

sin (\frac{180 x + 774 0 ^{\circ}}{180}) \cdot 3 - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180}) = 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

sin (\frac{180 x + 774 0 ^{\circ}}{180}) \cdot 3 - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180})

2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180}) = 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180})

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

:فعّل قانون القوى

cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180}) = cos^{1 + 1} (\frac{180 x + 774 0 ^{\circ}}{180})

= 2 cos^{1 + 1} (\frac{180 x + 774 0 ^{\circ}}{180})

1 + 1 = 2 :

اجمع الأعداد

= 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

= 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

= \frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) - 2 cos ^{2} ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

\frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) - 2 cos ^{2} ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} = 0

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

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

للطرفين

2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

أضف

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

ربّع الطرفين

(3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} = (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

من الطرفين

(2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

اطرح

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

حلل إلى عوامل:

(3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

(3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

كـ

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

اكتب مجددًا

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

3^{2}

كـ

9

اكتب مجددًا

= 3^{2} sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

2^{2}

كـ

4

اكتب مجددًا

= 3^{2} sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 2^{2} cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

a^{b c} = (a^{b})^{c}

:فعّل قانون القوى

cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180}) = (cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

= 3^{2} sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 2^{2} (cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

a^{m} b^{m} = (ab)^{m}

:فعّل قانون القوى

3^{2} sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

= (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - 2^{2} (cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

a^{m} b^{m} = (ab)^{m}

:فعّل قانون القوى

2^{2} (cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} = (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

= (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

= (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

x^{2} - y^{2} = (x + y) (x - y)

فعّل قانون فرق المربّعات

(3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} = (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))

= (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))

(3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) = 0

حلّ كل جزء على حدة

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0 or 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0 : x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

Rewrite using trig identities

2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180})

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

:فعّل نطريّة فيتاغوروس

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

= 2 (1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180})

(1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) \cdot 2 + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

بالاستعانة بطريقة التعويض

(1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) \cdot 2 + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = u :

على افتراض أنّ

(1 - u^{2}) \cdot 2 + 3 u = 0

(1 - u^{2}) \cdot 2 + 3 u = 0 : u = - \frac{1}{2}, u = 2

(1 - u^{2}) \cdot 2 + 3 u = 0

(1 - u^{2}) \cdot 2 + 3 u

وسّع:

2 - 2 u^{2} + 3 u

(1 - u^{2}) \cdot 2 + 3 u

= 2 (1 - u^{2}) + 3 u

2 (1 - u^{2})

وسٌع:

2 - 2 u^{2}

2 (1 - u^{2})

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = 2, b = 1, c = u^{2}

= 2 \cdot 1 - 2 u^{2}

2 \cdot 1 = 2 :

اضرب الأعداد

= 2 - 2 u^{2}

= 2 - 2 u^{2} + 3 u

2 - 2 u^{2} + 3 u = 0

a x^{2} + b x + c = 0

اكتب بالصورة الاعتياديّة

- 2 u^{2} + 3 u + 2 = 0

حلّ بالاستعانة بالصيغة التربيعيّة

- 2 u^{2} + 3 u + 2 = 0

الصيغة لحلّ المعادلة التربيعيّة

a = - 2, b = 3, c = 2

لـ

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 2 ) \cdot 2}{2 ( - 2 )}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 2 ) \cdot 2}{2 ( - 2 )}

3^{2} - 4 (- 2) \cdot 2 = 5

3^{2} - 4 (- 2) \cdot 2

- (- a) = a

فعّل القانون

= 3^{2} + 4 \cdot 2 \cdot 2

4 \cdot 2 \cdot 2 = 16 :

اضرب الأعداد

= 3^{2} + 16

3^{2} = 9

= 9 + 16

9 + 16 = 25 :

اجمع الأعداد

= 25

25 = 5^{2} :

حلّل العدد لعوامله أوّليّة

= 5^{2}

n a^{n} = a

:فعْل قانون الجذور

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 ( - 2 )}

Separate the solutions

u_{1} = \frac{- 3 + 5}{2 ( - 2 )}, u_{2} = \frac{- 3 - 5}{2 ( - 2 )}

u = \frac{- 3 + 5}{2 ( - 2 )} : - \frac{1}{2}

\frac{- 3 + 5}{2 ( - 2 )}

(- a) = - a

:احذف الأقواس

= \frac{- 3 + 5}{- 2 \cdot 2}

- 3 + 5 = 2 :

اطرح/اجمع الأعداد

= \frac{2}{- 2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{2}{- 4}

\frac{a}{- b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{2}{4}

2 :

إلغ العوامل المشتركة

= - \frac{1}{2}

u = \frac{- 3 - 5}{2 ( - 2 )} : 2

\frac{- 3 - 5}{2 ( - 2 )}

(- a) = - a

:احذف الأقواس

= \frac{- 3 - 5}{- 2 \cdot 2}

- 3 - 5 = - 8 :

اطرح الأعداد

= \frac{- 8}{- 2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{- 8}{- 4}

\frac{- a}{- b} = \frac{a}{b}

: استخدم ميزات الكسور التالية

= \frac{8}{4}

\frac{8}{4} = 2 :

اقسم الأعداد

= 2

حلول المعادلة التربيعيّة هي

u = - \frac{1}{2}, u = 2

u = sin (\frac{180 x + 774 0 ^{\circ}}{180})

استبدل مجددًا

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2}, sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2}, sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2} : x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2}

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2} :

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}

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n, \frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n, \frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n

حلّ:

x = 16 7^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n

180

اضرب الطرفين بـ

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n

180

اضرب الطرفين بـ

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 21 0^{\circ} + 180 \cdot 36 0^{\circ} n

بسّط

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 21 0^{\circ} + 180 \cdot 36 0^{\circ} n

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

بسّط:

180 x + 774 0^{\circ}

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

\frac{180}{180} = 1 :

اقسم الأعداد

= 180 x + 774 0^{\circ}

180 \cdot 21 0^{\circ} + 180 \cdot 36 0^{\circ} n

بسّط:

3780 0^{\circ} + 6480 0^{\circ} n

180 \cdot 21 0^{\circ} + 180 \cdot 36 0^{\circ} n

180 \cdot 21 0^{\circ} = 3780 0^{\circ}

180 \cdot 21 0^{\circ}

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

:اضرب كسور

= 3780 0^{\circ}

7 \cdot 180 = 1260 :

اضرب الأعداد

= 3780 0^{\circ}

\frac{1260}{6} = 210 :

اقسم الأعداد

= 3780 0^{\circ}

180 \cdot 36 0^{\circ} n = 6480 0^{\circ} n

180 \cdot 36 0^{\circ} n

180 \cdot 2 = 360 :

اضرب الأعداد

= 6480 0^{\circ} n

= 3780 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n

انقل

774 0^{\circ}

إلى الجانب الأيمن

180 x + 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n

من الطرفين

774 0^{\circ}

اطرح

180 x + 774 0^{\circ} - 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n - 774 0^{\circ}

بسّط

180 x = 3006 0^{\circ} + 6480 0^{\circ} n

180 x = 3006 0^{\circ} + 6480 0^{\circ} n

180

اقسم الطرفين على

180 x = 3006 0^{\circ} + 6480 0^{\circ} n

180

اقسم الطرفين على

\frac{180 x}{180} = 16 7^{\circ} + \frac{6480 0 ^{\circ} n}{180}

بسّط

x = 16 7^{\circ} + 36 0^{\circ} n

x = 16 7^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n

حلّ:

x = 28 7^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n

180

اضرب الطرفين بـ

\frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n

180

اضرب الطرفين بـ

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 33 0^{\circ} + 180 \cdot 36 0^{\circ} n

بسّط

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 33 0^{\circ} + 180 \cdot 36 0^{\circ} n

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

بسّط:

180 x + 774 0^{\circ}

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

\frac{180}{180} = 1 :

اقسم الأعداد

= 180 x + 774 0^{\circ}

180 \cdot 33 0^{\circ} + 180 \cdot 36 0^{\circ} n

بسّط:

5940 0^{\circ} + 6480 0^{\circ} n

180 \cdot 33 0^{\circ} + 180 \cdot 36 0^{\circ} n

180 \cdot 33 0^{\circ} = 5940 0^{\circ}

180 \cdot 33 0^{\circ}

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

:اضرب كسور

= 5940 0^{\circ}

11 \cdot 180 = 1980 :

اضرب الأعداد

= 5940 0^{\circ}

\frac{1980}{6} = 330 :

اقسم الأعداد

= 5940 0^{\circ}

180 \cdot 36 0^{\circ} n = 6480 0^{\circ} n

180 \cdot 36 0^{\circ} n

180 \cdot 2 = 360 :

اضرب الأعداد

= 6480 0^{\circ} n

= 5940 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n

انقل

774 0^{\circ}

إلى الجانب الأيمن

180 x + 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n

من الطرفين

774 0^{\circ}

اطرح

180 x + 774 0^{\circ} - 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n - 774 0^{\circ}

بسّط

180 x = 5166 0^{\circ} + 6480 0^{\circ} n

180 x = 5166 0^{\circ} + 6480 0^{\circ} n

180

اقسم الطرفين على

180 x = 5166 0^{\circ} + 6480 0^{\circ} n

180

اقسم الطرفين على

\frac{180 x}{180} = 28 7^{\circ} + \frac{6480 0 ^{\circ} n}{180}

بسّط

x = 28 7^{\circ} + 36 0^{\circ} n

x = 28 7^{\circ} + 36 0^{\circ} n

x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2 :

لا يوجد حلّ

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2

- 1 \leq sin (x) \leq 1

لا يوجد حلّ

وحّد الحلول

x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0 : x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

Rewrite using trig identities

- 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180})

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

:فعّل نطريّة فيتاغوروس

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

= - 2 (1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180})

- (1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) \cdot 2 + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

بالاستعانة بطريقة التعويض

- (1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) \cdot 2 + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = u :

على افتراض أنّ

- (1 - u^{2}) \cdot 2 + 3 u = 0

- (1 - u^{2}) \cdot 2 + 3 u = 0 : u = \frac{1}{2}, u = - 2

- (1 - u^{2}) \cdot 2 + 3 u = 0

- (1 - u^{2}) \cdot 2 + 3 u

وسّع:

- 2 + 2 u^{2} + 3 u

- (1 - u^{2}) \cdot 2 + 3 u

= - 2 (1 - u^{2}) + 3 u

- 2 (1 - u^{2})

وسٌع:

- 2 + 2 u^{2}

- 2 (1 - u^{2})

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = - 2, b = 1, c = u^{2}

= - 2 \cdot 1 - (- 2) u^{2}

فعّل قوانين سالب-موجب

- (- a) = a

= - 2 \cdot 1 + 2 u^{2}

2 \cdot 1 = 2 :

اضرب الأعداد

= - 2 + 2 u^{2}

= - 2 + 2 u^{2} + 3 u

- 2 + 2 u^{2} + 3 u = 0

a x^{2} + b x + c = 0

اكتب بالصورة الاعتياديّة

2 u^{2} + 3 u - 2 = 0

حلّ بالاستعانة بالصيغة التربيعيّة

2 u^{2} + 3 u - 2 = 0

الصيغة لحلّ المعادلة التربيعيّة

a = 2, b = 3, c = - 2

لـ

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

3^{2} - 4 \cdot 2 (- 2) = 5

3^{2} - 4 \cdot 2 (- 2)

- (- a) = a

فعّل القانون

= 3^{2} + 4 \cdot 2 \cdot 2

4 \cdot 2 \cdot 2 = 16 :

اضرب الأعداد

= 3^{2} + 16

3^{2} = 9

= 9 + 16

9 + 16 = 25 :

اجمع الأعداد

= 25

25 = 5^{2} :

حلّل العدد لعوامله أوّليّة

= 5^{2}

n a^{n} = a

:فعْل قانون الجذور

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 3 + 5}{2 \cdot 2}, u_{2} = \frac{- 3 - 5}{2 \cdot 2}

u = \frac{- 3 + 5}{2 \cdot 2} : \frac{1}{2}

\frac{- 3 + 5}{2 \cdot 2}

- 3 + 5 = 2 :

اطرح/اجمع الأعداد

= \frac{2}{2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{2}{4}

2 :

إلغ العوامل المشتركة

= \frac{1}{2}

u = \frac{- 3 - 5}{2 \cdot 2} : - 2

\frac{- 3 - 5}{2 \cdot 2}

- 3 - 5 = - 8 :

اطرح الأعداد

= \frac{- 8}{2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{- 8}{4}

\frac{- a}{b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{8}{4}

\frac{8}{4} = 2 :

اقسم الأعداد

= - 2

حلول المعادلة التربيعيّة هي

u = \frac{1}{2}, u = - 2

u = sin (\frac{180 x + 774 0 ^{\circ}}{180})

استبدل مجددًا

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2}, sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - 2

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2}, sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - 2

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2} : x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2}

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2} :

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}

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n, \frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n, \frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n

حلّ:

x = - 1 3^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n

180

اضرب الطرفين بـ

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n

180

اضرب الطرفين بـ

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 3 0^{\circ} + 180 \cdot 36 0^{\circ} n

بسّط

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 3 0^{\circ} + 180 \cdot 36 0^{\circ} n

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

بسّط:

180 x + 774 0^{\circ}

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

\frac{180}{180} = 1 :

اقسم الأعداد

= 180 x + 774 0^{\circ}

180 \cdot 3 0^{\circ} + 180 \cdot 36 0^{\circ} n

بسّط:

540 0^{\circ} + 6480 0^{\circ} n

180 \cdot 3 0^{\circ} + 180 \cdot 36 0^{\circ} n

180 \cdot 3 0^{\circ} = 540 0^{\circ}

180 \cdot 3 0^{\circ}

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

:اضرب كسور

= 540 0^{\circ}

\frac{180}{6} = 30 :

اقسم الأعداد

= 540 0^{\circ}

180 \cdot 36 0^{\circ} n = 6480 0^{\circ} n

180 \cdot 36 0^{\circ} n

180 \cdot 2 = 360 :

اضرب الأعداد

= 6480 0^{\circ} n

= 540 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n

انقل

774 0^{\circ}

إلى الجانب الأيمن

180 x + 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n

من الطرفين

774 0^{\circ}

اطرح

180 x + 774 0^{\circ} - 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n - 774 0^{\circ}

بسّط

180 x = - 234 0^{\circ} + 6480 0^{\circ} n

180 x = - 234 0^{\circ} + 6480 0^{\circ} n

180

اقسم الطرفين على

180 x = - 234 0^{\circ} + 6480 0^{\circ} n

180

اقسم الطرفين على

\frac{180 x}{180} = - 1 3^{\circ} + \frac{6480 0 ^{\circ} n}{180}

بسّط

x = - 1 3^{\circ} + 36 0^{\circ} n

x = - 1 3^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n

حلّ:

x = 10 7^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n

180

اضرب الطرفين بـ

\frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n

180

اضرب الطرفين بـ

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 15 0^{\circ} + 180 \cdot 36 0^{\circ} n

بسّط

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 15 0^{\circ} + 180 \cdot 36 0^{\circ} n

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

بسّط:

180 x + 774 0^{\circ}

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

\frac{180}{180} = 1 :

اقسم الأعداد

= 180 x + 774 0^{\circ}

180 \cdot 15 0^{\circ} + 180 \cdot 36 0^{\circ} n

بسّط:

2700 0^{\circ} + 6480 0^{\circ} n

180 \cdot 15 0^{\circ} + 180 \cdot 36 0^{\circ} n

180 \cdot 15 0^{\circ} = 2700 0^{\circ}

180 \cdot 15 0^{\circ}

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

:اضرب كسور

= 2700 0^{\circ}

5 \cdot 180 = 900 :

اضرب الأعداد

= 2700 0^{\circ}

\frac{900}{6} = 150 :

اقسم الأعداد

= 2700 0^{\circ}

180 \cdot 36 0^{\circ} n = 6480 0^{\circ} n

180 \cdot 36 0^{\circ} n

180 \cdot 2 = 360 :

اضرب الأعداد

= 6480 0^{\circ} n

= 2700 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n

انقل

774 0^{\circ}

إلى الجانب الأيمن

180 x + 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n

من الطرفين

774 0^{\circ}

اطرح

180 x + 774 0^{\circ} - 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n - 774 0^{\circ}

بسّط

180 x = 1926 0^{\circ} + 6480 0^{\circ} n

180 x = 1926 0^{\circ} + 6480 0^{\circ} n

180

اقسم الطرفين على

180 x = 1926 0^{\circ} + 6480 0^{\circ} n

180

اقسم الطرفين على

\frac{180 x}{180} = 10 7^{\circ} + \frac{6480 0 ^{\circ} n}{180}

بسّط

x = 10 7^{\circ} + 36 0^{\circ} n

x = 10 7^{\circ} + 36 0^{\circ} n

x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - 2 :

لا يوجد حلّ

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - 2

- 1 \leq sin (x) \leq 1

لا يوجد حلّ

وحّد الحلول

x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

وحّد الحلول

x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n, x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

تأكّد من صحّة الحلول عن طريق تعويضها في المعادلة الأصليّة

للتحقّق من دقّة الحلول

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

عوّض الحلول في

إلغي الحلول التي تعطي قضيّة كذب

16 7^{\circ} + 36 0^{\circ} n

افحص الحل:

خطأ

16 7^{\circ} + 36 0^{\circ} n

n = 1

استبدل

16 7^{\circ} + 36 0^{\circ} 1

x = 16 7^{\circ} + 36 0^{\circ} 1

عوّض

, 3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

في

3 tan (16 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ}) = 2 cos (16 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ})

بسّط

1.73205 \dots = - 1.73205 \dots

\Rightarrow خطأ

28 7^{\circ} + 36 0^{\circ} n

افحص الحل:

خطأ

28 7^{\circ} + 36 0^{\circ} n

n = 1

استبدل

28 7^{\circ} + 36 0^{\circ} 1

x = 28 7^{\circ} + 36 0^{\circ} 1

عوّض

, 3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

في

3 tan (28 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ}) = 2 cos (28 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ})

بسّط

- 1.73205 \dots = 1.73205 \dots

\Rightarrow خطأ

- 1 3^{\circ} + 36 0^{\circ} n

افحص الحل:

صحيح

- 1 3^{\circ} + 36 0^{\circ} n

n = 1

استبدل

- 1 3^{\circ} + 36 0^{\circ} 1

x = - 1 3^{\circ} + 36 0^{\circ} 1

عوّض

, 3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

في

3 tan (- 1 3^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ}) = 2 cos (- 1 3^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ})

بسّط

1.73205 \dots = 1.73205 \dots

\Rightarrow صحيح

10 7^{\circ} + 36 0^{\circ} n

افحص الحل:

صحيح

10 7^{\circ} + 36 0^{\circ} n

n = 1

استبدل

10 7^{\circ} + 36 0^{\circ} 1

x = 10 7^{\circ} + 36 0^{\circ} 1

عوّض

, 3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

في

3 tan (10 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ}) = 2 cos (10 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ})

بسّط

- 1.73205 \dots = - 1.73205 \dots

\Rightarrow صحيح

x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

رسم

أمثلة شائعة

4sin(x)=sqrt(3)sec(x)

4 sin (x) = 3 sec (x)

(sin(55.01-x))/(sin(x))= 1200/500

\frac{sin ( 55.01 - x )}{sin ( x )} = \frac{1200}{500}

csc(θ)=0.77

csc (θ) = 0.77

(sin(x)+cos(x))/(sin(x))=1 1/(tan(x))

\frac{sin ( x ) + cos ( x )}{sin ( x )} = 1 \frac{1}{tan ( x )}

arctan(x)=(-pi)/2

arctan (x) = \frac{- π}{2}