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شائع علم المثلثات >

solvefor b,a=arccos((a^2-b^2-c^2)/(-2bc))

الحلّ

solve for

b, a = arccos (\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c})

الحلّ

b = c cos (a) + c^{2} cos^{2} (a) - c^{2} + a^{2}, b = c cos (a) - c^{2} cos^{2} (a) - c^{2} + a^{2}

خطوات الحلّ

a = arccos (\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c})

بدّل الأطراف

arccos (\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c}) = a

Apply trig inverse properties

arccos (\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c}) = a

arccos (x) = a \Rightarrow x = cos (a)

\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c} = cos (a)

\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c} = cos (a)

\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c} = cos (a)

حلّ:

b = c cos (a) + c^{2} cos^{2} (a) - c^{2} + a^{2}, b = c cos (a) - c^{2} cos^{2} (a) - c^{2} + a^{2}; c \neq = 0

\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c} = cos (a)

\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c}

بسّط:

- \frac{a ^{2} - b ^{2} - c ^{2}}{2 b c}

\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c}

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

: استخدم ميزات الكسور التالية

= - \frac{a ^{2} - b ^{2} - c ^{2}}{2 b c}

- \frac{a ^{2} - b ^{2} - c ^{2}}{2 b c} = cos (a)

b

اضرب الطرفين بـ

- \frac{a ^{2} - b ^{2} - c ^{2}}{2 b c} = cos (a)

b

اضرب الطرفين بـ

- \frac{a ^{2} - b ^{2} - c ^{2}}{2 b c} b = cos (a) b

بسّط

- \frac{a ^{2} - b ^{2} - c ^{2}}{2 c} = cos (a) b

- \frac{a ^{2} - b ^{2} - c ^{2}}{2 c} = cos (a) b

- \frac{a ^{2} - b ^{2} - c ^{2}}{2 c} = cos (a) b

حلّ:

b = c cos (a) + c^{2} cos^{2} (a) - c^{2} + a^{2}, b = c cos (a) - c^{2} cos^{2} (a) - c^{2} + a^{2}

- \frac{a ^{2} - b ^{2} - c ^{2}}{2 c} = cos (a) b

2 c

اضرب الطرفين بـ

- \frac{a ^{2} - b ^{2} - c ^{2}}{2 c} = cos (a) b

2 c

اضرب الطرفين بـ

- \frac{a ^{2} - b ^{2} - c ^{2}}{2 c} \cdot 2 c = cos (a) b \cdot 2 c; c \neq = 0

بسّط

- (a^{2} - b^{2} - c^{2}) = 2 b c cos (a); c \neq = 0

- (a^{2} - b^{2} - c^{2}) = 2 b c cos (a); c \neq = 0

- (a^{2} - b^{2} - c^{2})

وسّع:

- a^{2} + b^{2} + c^{2}

- (a^{2} - b^{2} - c^{2})

افتح أقواس

= - a^{2} - (- b^{2}) - (- c^{2})

فعّل قوانين سالب-موجب

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

= - a^{2} + b^{2} + c^{2}

- a^{2} + b^{2} + c^{2} = 2 b c cos (a); c \neq = 0

انقل

2 b c cos (a)

إلى الجانب الأيسر

- a^{2} + b^{2} + c^{2} = 2 b c cos (a); c \neq = 0

من الطرفين

2 b c cos (a)

اطرح

- a^{2} + b^{2} + c^{2} - 2 b c cos (a) = 2 b c cos (a) - 2 b c cos (a); c \neq = 0

بسّط

- a^{2} + b^{2} + c^{2} - 2 b c cos (a) = 0; c \neq = 0

- a^{2} + b^{2} + c^{2} - 2 b c cos (a) = 0; c \neq = 0

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

اكتب بالصورة الاعتياديّة

b^{2} - 2 c cos (a) b - a^{2} + c^{2} = 0; c \neq = 0

حلّ بالاستعانة بالصيغة التربيعيّة

b^{2} - 2 c cos (a) b - a^{2} + c^{2} = 0

الصيغة لحلّ المعادلة التربيعيّة

a = 1, b = - 2 c cos (a), c = - a^{2} + c^{2}

لـ

b_{1, 2} = \frac{- ( - 2 c cos ( a ) ) \pm ( - 2 c cos ( a ) ) ^{2} - 4 \cdot 1 \cdot ( - a ^{2} + c ^{2} )}{2 \cdot 1}

b_{1, 2} = \frac{- ( - 2 c cos ( a ) ) \pm ( - 2 c cos ( a ) ) ^{2} - 4 \cdot 1 \cdot ( - a ^{2} + c ^{2} )}{2 \cdot 1}

(- 2 c cos (a))^{2} - 4 \cdot 1 \cdot (- a^{2} + c^{2})

بسّط:

2 c^{2} cos^{2} (a) + a^{2} - c^{2}

(- 2 c cos (a))^{2} - 4 \cdot 1 \cdot (- a^{2} + c^{2})

(- 2 c cos (a))^{2} = 2^{2} c^{2} cos^{2} (a)

(- 2 c cos (a))^{2}

زوجيّ n

إذا تحقّق أنّ

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

:فعّل قانون القوى

(- 2 c cos (a))^{2} = (2 c cos (a))^{2}

= (2 c cos (a))^{2}

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

:فعّل قانون القوى

= 2^{2} c^{2} cos^{2} (a)

4 \cdot 1 \cdot (- a^{2} + c^{2}) = 4 (- a^{2} + c^{2})

4 \cdot 1 \cdot (- a^{2} + c^{2})

4 \cdot 1 = 4 :

اضرب الأعداد

= 4 (c^{2} - a^{2})

= 2^{2} c^{2} cos^{2} (a) - 4 (c^{2} - a^{2})

2^{2} c^{2} cos^{2} (a) - 4 (- a^{2} + c^{2})

حلل إلى عوامل:

4 (c^{2} cos^{2} (a) + a^{2} - c^{2})

2^{2} c^{2} cos^{2} (a) - 4 (- a^{2} + c^{2})

أعد الكتابة كـ

= 4 c^{2} cos^{2} (a) - 4 (- a^{2} + c^{2})

4

قم باخراج العامل المشترك

= 4 (c^{2} cos^{2} (a) - (- a^{2} + c^{2}))

c^{2} cos^{2} (a) - (c^{2} - a^{2})

وسٌع:

c^{2} cos^{2} (a) + a^{2} - c^{2}

c^{2} cos^{2} (a) - (- a^{2} + c^{2})

- (- a^{2} + c^{2}) : a^{2} - c^{2}

- (- a^{2} + c^{2})

افتح أقواس

= - (- a^{2}) - c^{2}

فعّل قوانين سالب-موجب

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

= a^{2} - c^{2}

= c^{2} cos^{2} (a) + a^{2} - c^{2}

= 4 (c^{2} cos^{2} (a) - c^{2} + a^{2})

= 4 (c^{2} cos^{2} (a) + a^{2} - c^{2})

a \geq 0, b \geq 0

بافتراض أنّ

n ab = n a n b

:فعّل قانون الجذور

= 4 c^{2} cos^{2} (a) - c^{2} + a^{2}

4 = 2

4

4 = 2^{2} :

حلّل العدد لعوامله أوّليّة

= 2^{2}

n a^{n} = a

:فعْل قانون الجذور

2^{2} = 2

= 2

= 2 c^{2} cos^{2} (a) - c^{2} + a^{2}

b_{1, 2} = \frac{- ( - 2 c cos ( a ) ) \pm 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1}

Separate the solutions

b_{1} = \frac{- ( - 2 c cos ( a ) ) + 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1}, b_{2} = \frac{- ( - 2 c cos ( a ) ) - 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1}

b = \frac{- ( - 2 c cos ( a ) ) + 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1} : c cos (a) + c^{2} cos^{2} (a) - c^{2} + a^{2}

\frac{- ( - 2 c cos ( a ) ) + 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1}

- (- a) = a

فعّل القانون

= \frac{2 c cos ( a ) + 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1}

2 \cdot 1 = 2 :

اضرب الأعداد

= \frac{2 c cos ( a ) + 2 c ^{2} cos ^{2} ( a ) - c ^{2} + a ^{2}}{2}

2

قم باخراج العامل المشترك

= \frac{2 ( c cos ( a ) + a ^{2} - c ^{2} + cos ^{2} ( a ) c ^{2} )}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= c cos (a) + c^{2} cos^{2} (a) - c^{2} + a^{2}

b = \frac{- ( - 2 c cos ( a ) ) - 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1} : c cos (a) - c^{2} cos^{2} (a) - c^{2} + a^{2}

\frac{- ( - 2 c cos ( a ) ) - 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1}

- (- a) = a

فعّل القانون

= \frac{2 c cos ( a ) - 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1}

2 \cdot 1 = 2 :

اضرب الأعداد

= \frac{2 c cos ( a ) - 2 c ^{2} cos ^{2} ( a ) - c ^{2} + a ^{2}}{2}

2

قم باخراج العامل المشترك

= \frac{2 ( c cos ( a ) - a ^{2} - c ^{2} + cos ^{2} ( a ) c ^{2} )}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= c cos (a) - c^{2} cos^{2} (a) - c^{2} + a^{2}

حلول المعادلة التربيعيّة هي

b = c cos (a) + c^{2} cos^{2} (a) - c^{2} + a^{2}, b = c cos (a) - c^{2} cos^{2} (a) - c^{2} + a^{2}

b = c cos (a) + c^{2} cos^{2} (a) - c^{2} + a^{2}, b = c cos (a) - c^{2} cos^{2} (a) - c^{2} + a^{2}; c \neq = 0

b = c cos (a) + c^{2} cos^{2} (a) - c^{2} + a^{2}, b = c cos (a) - c^{2} cos^{2} (a) - c^{2} + a^{2}

رسم

أمثلة شائعة

4cos(θ)-1=0

4 cos (θ) - 1 = 0

sec(y)+5tan(y)=3cos(y)

sec (y) + 5 tan (y) = 3 cos (y)

-2sinh(t-2)=0

- 2 sinh (t - 2) = 0

cos(x)sin(x)+2cos^2(x)=0

cos (x) sin (x) + 2 cos^{2} (x) = 0

tan(x)= 130/45

tan (x) = \frac{130}{45}