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شائع علم المثلثات >

9.8sin(α)-1.96cos(α)=0.47

الحلّ

9.8 \cdot sin (α) - 1.96 \cdot cos (α) = 0.47

الحلّ

α = - 2.99124 \dots + 2 πn, α = 0.24444 \dots + 2 πn

درجات

α = - 171.38555 \dots^{\circ} + 36 0^{\circ} n, α = 14.00542 \dots^{\circ} + 36 0^{\circ} n

خطوات الحلّ

9.8 sin (α) - 1.96 cos (α) = 0.47

للطرفين

1.96 cos (α)

أضف

9.8 sin (α) = 0.47 + 1.96 cos (α)

ربّع الطرفين

(9.8 sin (α))^{2} = (0.47 + 1.96 cos (α))^{2}

من الطرفين

(0.47 + 1.96 cos (α))^{2}

اطرح

96.04 sin^{2} (α) - 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) = 0

Rewrite using trig identities

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 sin^{2} (α)

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

:فعّل نطريّة فيتاغوروس

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

= - 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 (1 - cos^{2} (α))

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 (1 - cos^{2} (α))

بسّط:

- 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 (1 - cos^{2} (α))

96.04 (1 - cos^{2} (α))

وسٌع:

96.04 - 96.04 cos^{2} (α)

96.04 (1 - cos^{2} (α))

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = 96.04, b = 1, c = cos^{2} (α)

= 96.04 \cdot 1 - 96.04 cos^{2} (α)

= 1 \cdot 96.04 - 96.04 cos^{2} (α)

1 \cdot 96.04 = 96.04 :

اضرب الأعداد

= 96.04 - 96.04 cos^{2} (α)

= - 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 - 96.04 cos^{2} (α)

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 - 96.04 cos^{2} (α)

بسّط:

- 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 - 96.04 cos^{2} (α)

جمّع التعابير المتشابهة

= - 1.8424 cos (α) - 3.8416 cos^{2} (α) - 96.04 cos^{2} (α) - 0.2209 + 96.04

- 3.8416 cos^{2} (α) - 96.04 cos^{2} (α) = - 99.8816 cos^{2} (α) :

اجمع العناصر المتشابهة

= - 1.8424 cos (α) - 99.8816 cos^{2} (α) - 0.2209 + 96.04

- 0.2209 + 96.04 = 95.8191 :

اطرح/اجمع الأعداد

= - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

= - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

= - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

95.8191 - 1.8424 cos (α) - 99.8816 cos^{2} (α) = 0

بالاستعانة بطريقة التعويض

95.8191 - 1.8424 cos (α) - 99.8816 cos^{2} (α) = 0

cos (α) = u :

على افتراض أنّ

95.8191 - 1.8424 u - 99.8816 u^{2} = 0

95.8191 - 1.8424 u - 99.8816 u^{2} = 0 : u = - \frac{1.8424 + 38285.654512}{199.7632}, u = \frac{38285.654512 - 1.8424}{199.7632}

95.8191 - 1.8424 u - 99.8816 u^{2} = 0

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

اكتب بالصورة الاعتياديّة

- 99.8816 u^{2} - 1.8424 u + 95.8191 = 0

حلّ بالاستعانة بالصيغة التربيعيّة

- 99.8816 u^{2} - 1.8424 u + 95.8191 = 0

الصيغة لحلّ المعادلة التربيعيّة

a = - 99.8816, b = - 1.8424, c = 95.8191

لـ

u_{1, 2} = \frac{- ( - 1.8424 ) \pm ( - 1.8424 ) ^{2} - 4 ( - 99.8816 ) \cdot 95.8191}{2 ( - 99.8816 )}

u_{1, 2} = \frac{- ( - 1.8424 ) \pm ( - 1.8424 ) ^{2} - 4 ( - 99.8816 ) \cdot 95.8191}{2 ( - 99.8816 )}

(- 1.8424)^{2} - 4 (- 99.8816) \cdot 95.8191 = 38285.654512

(- 1.8424)^{2} - 4 (- 99.8816) \cdot 95.8191

- (- a) = a

فعّل القانون

= (- 1.8424)^{2} + 4 \cdot 99.8816 \cdot 95.8191

زوجيّ n

إذا تحقّق أنّ

, (- a)^{n} = a^{n}

:فعّل قانون القوى

(- 1.8424)^{2} = 1.842 4^{2}

= 1.842 4^{2} + 4 \cdot 95.8191 \cdot 99.8816

4 \cdot 99.8816 \cdot 95.8191 = 38282.26007 \dots :

اضرب الأعداد

= 1.842 4^{2} + 38282.26007 \dots

1.842 4^{2} = 3.39443776

= 3.39443776 + 38282.26007 \dots

3.39443776 + 38282.26007 \dots = 38285.654512 :

اجمع الأعداد

= 38285.654512

u_{1, 2} = \frac{- ( - 1.8424 ) \pm 38285.654512}{2 ( - 99.8816 )}

Separate the solutions

u_{1} = \frac{- ( - 1.8424 ) + 38285.654512}{2 ( - 99.8816 )}, u_{2} = \frac{- ( - 1.8424 ) - 38285.654512}{2 ( - 99.8816 )}

u = \frac{- ( - 1.8424 ) + 38285.654512}{2 ( - 99.8816 )} : - \frac{1.8424 + 38285.654512}{199.7632}

\frac{- ( - 1.8424 ) + 38285.654512}{2 ( - 99.8816 )}

(- a) = - a, - (- a) = a

:احذف الأقواس

= \frac{1.8424 + 38285.654512}{- 2 \cdot 99.8816}

2 \cdot 99.8816 = 199.7632 :

اضرب الأعداد

= \frac{1.8424 + 38285.654512}{- 199.7632}

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

: استخدم ميزات الكسور التالية

= - \frac{1.8424 + 38285.654512}{199.7632}

u = \frac{- ( - 1.8424 ) - 38285.654512}{2 ( - 99.8816 )} : \frac{38285.654512 - 1.8424}{199.7632}

\frac{- ( - 1.8424 ) - 38285.654512}{2 ( - 99.8816 )}

(- a) = - a, - (- a) = a

:احذف الأقواس

= \frac{1.8424 - 38285.654512}{- 2 \cdot 99.8816}

2 \cdot 99.8816 = 199.7632 :

اضرب الأعداد

= \frac{1.8424 - 38285.654512}{- 199.7632}

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

: استخدم ميزات الكسور التالية

1.8424 - 38285.654512 = - (38285.654512 - 1.8424)

= \frac{38285.654512 - 1.8424}{199.7632}

حلول المعادلة التربيعيّة هي

u = - \frac{1.8424 + 38285.654512}{199.7632}, u = \frac{38285.654512 - 1.8424}{199.7632}

u = cos (α)

استبدل مجددًا

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}, cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}, cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632} : α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}

Apply trig inverse properties

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632} :

حلول عامّة لـ

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

cos (α) = \frac{38285.654512 - 1.8424}{199.7632} : α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

Apply trig inverse properties

cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

cos (α) = \frac{38285.654512 - 1.8424}{199.7632} :

حلول عامّة لـ

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

وحّد الحلول

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

تأكّد من صحّة الحلول عن طريق تعويضها في المعادلة الأصليّة

للتحقّق من دقّة الحلول

9.8 sin (α) - 1.96 cos (α) = 0.47

عوّض الحلول في

إلغي الحلول التي تعطي قضيّة كذب

arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

افحص الحل:

خطأ

arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

n = 1

استبدل

arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

عوّض

, 9.8 sin (α) - 1.96 cos (α) = 0.47

في

9.8 sin (arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) - 1.96 cos (arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) = 0.47

بسّط

3.40577 \dots = 0.47

\Rightarrow خطأ

- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

افحص الحل:

صحيح

- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

n = 1

استبدل

- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

عوّض

, 9.8 sin (α) - 1.96 cos (α) = 0.47

في

9.8 sin (- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) - 1.96 cos (- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) = 0.47

بسّط

0.47 = 0.47

\Rightarrow صحيح

arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

افحص الحل:

صحيح

arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

n = 1

استبدل

arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

عوّض

, 9.8 sin (α) - 1.96 cos (α) = 0.47

في

9.8 sin (arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) - 1.96 cos (arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) = 0.47

بسّط

0.47 = 0.47

\Rightarrow صحيح

2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

افحص الحل:

خطأ

2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

n = 1

استبدل

2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

عوّض

, 9.8 sin (α) - 1.96 cos (α) = 0.47

في

9.8 sin (2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) - 1.96 cos (2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) = 0.47

بسّط

- 4.27346 \dots = 0.47

\Rightarrow خطأ

α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

أظهر الحلّ بالتمثيل العشريّ

α = - 2.99124 \dots + 2 πn, α = 0.24444 \dots + 2 πn

رسم

أمثلة شائعة

cos(pi+x)= 1/(sqrt(2))

2sin(x)=sec(x)

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

2sec(x)=tan(x)+cot(x)

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

9.8*sin(α)-1.96*cos(α)=0.47

الحلّ

الحلّ

رسم

أمثلة شائعة

9.8sin(α)-1.96cos(α)=0.47