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شائع علم المثلثات >

88.2sin(x)-12.78=0.1*88.2cos(x)

الحلّ

88.2 sin (x) - 12.78 = 0.1 \cdot 88.2 cos (x)

الحلّ

x = 0.24435 \dots + 2 πn, x = π - 0.04501 \dots + 2 πn

درجات

x = 14.00032 \dots^{\circ} + 36 0^{\circ} n, x = 177.42086 \dots^{\circ} + 36 0^{\circ} n

خطوات الحلّ

88.2 sin (x) - 12.78 = 0.1 \cdot 88.2 cos (x)

ربّع الطرفين

(88.2 sin (x) - 12.78)^{2} = (0.1 \cdot 88.2 cos (x))^{2}

من الطرفين

(0.188.2 cos (x))^{2}

اطرح

(88.2 sin (x) - 12.78)^{2} - 77.7924 cos^{2} (x) = 0

Rewrite using trig identities

(- 12.78 + 88.2 sin (x))^{2} - 77.7924 cos^{2} (x)

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

:فعّل نطريّة فيتاغوروس

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

= (- 12.78 + 88.2 sin (x))^{2} - 77.7924 (1 - sin^{2} (x))

(- 12.78 + 88.2 sin (x))^{2} - 77.7924 (1 - sin^{2} (x))

بسّط:

7857.0324 sin^{2} (x) - 2254.392 sin (x) + 85.536

(- 12.78 + 88.2 sin (x))^{2} - 77.7924 (1 - sin^{2} (x))

(- 12.78 + 88.2 sin (x))^{2} : 163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x)

(a + b)^{2} = a^{2} + 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

a = - 12.78, b = 88.2 sin (x)

= (- 12.78)^{2} + 2 (- 12.78) \cdot 88.2 sin (x) + (88.2 sin (x))^{2}

(- 12.78)^{2} + 2 (- 12.78) \cdot 88.2 sin (x) + (88.2 sin (x))^{2}

بسّط:

163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x)

(- 12.78)^{2} + 2 (- 12.78) \cdot 88.2 sin (x) + (88.2 sin (x))^{2}

(- a) = - a

:احذف الأقواس

= (- 12.78)^{2} - 2 \cdot 12.78 \cdot 88.2 sin (x) + (88.2 sin (x))^{2}

(- 12.78)^{2} = 163.3284

(- 12.78)^{2}

زوجيّ n

إذا تحقّق أنّ

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

:فعّل قانون القوى

(- 12.78)^{2} = 12.7 8^{2}

= 12.7 8^{2}

12.7 8^{2} = 163.3284

= 163.3284

2 \cdot 12.78 \cdot 88.2 sin (x) = 2254.392 sin (x)

2 \cdot 12.78 \cdot 88.2 sin (x)

2 \cdot 12.78 \cdot 88.2 = 2254.392 :

اضرب الأعداد

= 2254.392 sin (x)

(88.2 sin (x))^{2} = 7779.24 sin^{2} (x)

(88.2 sin (x))^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 88. 2^{2} sin^{2} (x)

88. 2^{2} = 7779.24

= 7779.24 sin^{2} (x)

= 163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x)

= 163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x)

= 163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x) - 77.7924 (1 - sin^{2} (x))

- 77.7924 (1 - sin^{2} (x))

وسٌع:

- 77.7924 + 77.7924 sin^{2} (x)

- 77.7924 (1 - sin^{2} (x))

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = - 77.7924, b = 1, c = sin^{2} (x)

= - 77.7924 \cdot 1 - (- 77.7924) sin^{2} (x)

فعّل قوانين سالب-موجب

- (- a) = a

= - 1 \cdot 77.7924 + 77.7924 sin^{2} (x)

1 \cdot 77.7924 = 77.7924 :

اضرب الأعداد

= - 77.7924 + 77.7924 sin^{2} (x)

= 163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x) - 77.7924 + 77.7924 sin^{2} (x)

163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x) - 77.7924 + 77.7924 sin^{2} (x)

بسّط:

7857.0324 sin^{2} (x) - 2254.392 sin (x) + 85.536

163.3284 - 2254.392 sin (x) + 7779.24 sin^{2} (x) - 77.7924 + 77.7924 sin^{2} (x)

جمّع التعابير المتشابهة

= - 2254.392 sin (x) + 7779.24 sin^{2} (x) + 77.7924 sin^{2} (x) + 163.3284 - 77.7924

7779.24 sin^{2} (x) + 77.7924 sin^{2} (x) = 7857.0324 sin^{2} (x) :

اجمع العناصر المتشابهة

= - 2254.392 sin (x) + 7857.0324 sin^{2} (x) + 163.3284 - 77.7924

163.3284 - 77.7924 = 85.536 :

اطرح/اجمع الأعداد

= 7857.0324 sin^{2} (x) - 2254.392 sin (x) + 85.536

= 7857.0324 sin^{2} (x) - 2254.392 sin (x) + 85.536

= 7857.0324 sin^{2} (x) - 2254.392 sin (x) + 85.536

85.536 - 2254.392 sin (x) + 7857.0324 sin^{2} (x) = 0

بالاستعانة بطريقة التعويض

85.536 - 2254.392 sin (x) + 7857.0324 sin^{2} (x) = 0

sin (x) = u :

على افتراض أنّ

85.536 - 2254.392 u + 7857.0324 u^{2} = 0

85.536 - 2254.392 u + 7857.0324 u^{2} = 0 : u = \frac{0.28692 \dots + 0.03878 \dots}{2}, u = \frac{0.28692 \dots - 0.03878 \dots}{2}

85.536 - 2254.392 u + 7857.0324 u^{2} = 0

7857.0324

اقسم الطرفين على

\frac{85.536}{7857.0324} - \frac{2254.392 u}{7857.0324} + \frac{7857.0324 u ^{2}}{7857.0324} = \frac{0}{7857.0324}

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

اكتب بالصورة الاعتياديّة

u^{2} - 0.28692 \dots u + 0.01088 \dots = 0

حلّ بالاستعانة بالصيغة التربيعيّة

u^{2} - 0.28692 \dots u + 0.01088 \dots = 0

الصيغة لحلّ المعادلة التربيعيّة

a = 1, b = - 0.28692 \dots, c = 0.01088 \dots

لـ

u_{1, 2} = \frac{- ( - 0.28692 \dots ) \pm ( - 0.28692 \dots ) ^{2} - 4 \cdot 1 \cdot 0.01088 \dots}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 0.28692 \dots ) \pm ( - 0.28692 \dots ) ^{2} - 4 \cdot 1 \cdot 0.01088 \dots}{2 \cdot 1}

(- 0.28692 \dots)^{2} - 4 \cdot 1 \cdot 0.01088 \dots = 0.03878 \dots

(- 0.28692 \dots)^{2} - 4 \cdot 1 \cdot 0.01088 \dots

زوجيّ n

إذا تحقّق أنّ

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

:فعّل قانون القوى

(- 0.28692 \dots)^{2} = 0.28692 \dots^{2}

= 0.28692 \dots^{2} - 4 \cdot 1 \cdot 0.01088 \dots

4 \cdot 1 \cdot 0.01088 \dots = 0.04354 \dots :

اضرب الأعداد

= 0.28692 \dots^{2} - 0.04354 \dots

0.28692 \dots^{2} = 0.08232 \dots

= 0.08232 \dots - 0.04354 \dots

0.08232 \dots - 0.04354 \dots = 0.03878 \dots :

اطرح الأعداد

= 0.03878 \dots

u_{1, 2} = \frac{- ( - 0.28692 \dots ) \pm 0.03878 \dots}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 0.28692 \dots ) + 0.03878 \dots}{2 \cdot 1}, u_{2} = \frac{- ( - 0.28692 \dots ) - 0.03878 \dots}{2 \cdot 1}

u = \frac{- ( - 0.28692 \dots ) + 0.03878 \dots}{2 \cdot 1} : \frac{0.28692 \dots + 0.03878 \dots}{2}

\frac{- ( - 0.28692 \dots ) + 0.03878 \dots}{2 \cdot 1}

- (- a) = a

فعّل القانون

= \frac{0.28692 \dots + 0.03878 \dots}{2 \cdot 1}

2 \cdot 1 = 2 :

اضرب الأعداد

= \frac{0.28692 \dots + 0.03878 \dots}{2}

u = \frac{- ( - 0.28692 \dots ) - 0.03878 \dots}{2 \cdot 1} : \frac{0.28692 \dots - 0.03878 \dots}{2}

\frac{- ( - 0.28692 \dots ) - 0.03878 \dots}{2 \cdot 1}

- (- a) = a

فعّل القانون

= \frac{0.28692 \dots - 0.03878 \dots}{2 \cdot 1}

2 \cdot 1 = 2 :

اضرب الأعداد

= \frac{0.28692 \dots - 0.03878 \dots}{2}

حلول المعادلة التربيعيّة هي

u = \frac{0.28692 \dots + 0.03878 \dots}{2}, u = \frac{0.28692 \dots - 0.03878 \dots}{2}

u = sin (x)

استبدل مجددًا

sin (x) = \frac{0.28692 \dots + 0.03878 \dots}{2}, sin (x) = \frac{0.28692 \dots - 0.03878 \dots}{2}

sin (x) = \frac{0.28692 \dots + 0.03878 \dots}{2}, sin (x) = \frac{0.28692 \dots - 0.03878 \dots}{2}

sin (x) = \frac{0.28692 \dots + 0.03878 \dots}{2} : x = arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn

sin (x) = \frac{0.28692 \dots + 0.03878 \dots}{2}

Apply trig inverse properties

sin (x) = \frac{0.28692 \dots + 0.03878 \dots}{2}

sin (x) = \frac{0.28692 \dots + 0.03878 \dots}{2} :

حلول عامّة لـ

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn

x = arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn

sin (x) = \frac{0.28692 \dots - 0.03878 \dots}{2} : x = arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

sin (x) = \frac{0.28692 \dots - 0.03878 \dots}{2}

Apply trig inverse properties

sin (x) = \frac{0.28692 \dots - 0.03878 \dots}{2}

sin (x) = \frac{0.28692 \dots - 0.03878 \dots}{2} :

حلول عامّة لـ

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

x = arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

وحّد الحلول

x = arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn, x = arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

تأكّد من صحّة الحلول عن طريق تعويضها في المعادلة الأصليّة

للتحقّق من دقّة الحلول

88.2 sin (x) - 12.78 = 0.188.2 cos (x)

عوّض الحلول في

إلغي الحلول التي تعطي قضيّة كذب

arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn

افحص الحل:

صحيح

arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn

n = 1

استبدل

arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1

x = arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1

عوّض

, 88.2 sin (x) - 12.78 = 0.188.2 cos (x)

في

88.2 sin (arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1) - 12.78 = 0.1 \cdot 88.2 cos (arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1)

بسّط

8.55799 \dots = 8.55799 \dots

\Rightarrow صحيح

π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn

افحص الحل:

خطأ

π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn

n = 1

استبدل

π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1

x = π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1

عوّض

, 88.2 sin (x) - 12.78 = 0.188.2 cos (x)

في

88.2 sin (π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1) - 12.78 = 0.1 \cdot 88.2 cos (π - arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 π 1)

بسّط

8.55799 \dots = - 8.55799 \dots

\Rightarrow خطأ

arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

افحص الحل:

خطأ

arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

n = 1

استبدل

arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1

x = arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1

عوّض

, 88.2 sin (x) - 12.78 = 0.188.2 cos (x)

في

88.2 sin (arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1) - 12.78 = 0.1 \cdot 88.2 cos (arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1)

بسّط

- 8.81106 \dots = 8.81106 \dots

\Rightarrow خطأ

π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

افحص الحل:

صحيح

π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

n = 1

استبدل

π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1

x = π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1

عوّض

, 88.2 sin (x) - 12.78 = 0.188.2 cos (x)

في

88.2 sin (π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1) - 12.78 = 0.1 \cdot 88.2 cos (π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 π 1)

بسّط

- 8.81106 \dots = - 8.81106 \dots

\Rightarrow صحيح

x = arcsin (\frac{0.28692 \dots + 0.03878 \dots}{2}) + 2 πn, x = π - arcsin (\frac{0.28692 \dots - 0.03878 \dots}{2}) + 2 πn

أظهر الحلّ بالتمثيل العشريّ

x = 0.24435 \dots + 2 πn, x = π - 0.04501 \dots + 2 πn

رسم

أمثلة شائعة

1+sin(x)=2*cos(x)

1 + sin (x) = 2 \cdot cos (x)

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tan(x)=1.15

tan (x) = 1.15

0.6=cos^2(x)

0.6 = cos^{2} (x)

10=sqrt(65)*sqrt(5)*cos(θ)

10 = 65 \cdot 5 \cdot cos (θ)