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شائع علم المثلثات >

3tan(x)+cot(x)-5csc(x)=0

الحلّ

3 tan (x) + cot (x) - 5 csc (x) = 0

الحلّ

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

درجات

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

3 tan (x) + cot (x) - 5 csc (x) = 0

sin, cos :

عبّر بواسطة

cot (x) + 3 tan (x) - 5 csc (x)

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

:Use the basic trigonometric identity

= \frac{cos ( x )}{sin ( x )} + 3 tan (x) - 5 csc (x)

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

:Use the basic trigonometric identity

= \frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{sin ( x )}{cos ( x )} - 5 csc (x)

csc (x) = \frac{1}{sin ( x )}

:Use the basic trigonometric identity

= \frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{sin ( x )}{cos ( x )} - 5 \cdot \frac{1}{sin ( x )}

\frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{sin ( x )}{cos ( x )} - 5 \cdot \frac{1}{sin ( x )}

بسّط:

\frac{cos ( x ) ( cos ( x ) - 5 ) + 3 sin ^{2} ( x )}{sin ( x ) cos ( x )}

\frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{sin ( x )}{cos ( x )} - 5 \cdot \frac{1}{sin ( x )}

3 \cdot \frac{sin ( x )}{cos ( x )} = \frac{3 sin ( x )}{cos ( x )}

3 \cdot \frac{sin ( x )}{cos ( x )}

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

:اضرب كسور

= \frac{sin ( x ) \cdot 3}{cos ( x )}

5 \cdot \frac{1}{sin ( x )} = \frac{5}{sin ( x )}

5 \cdot \frac{1}{sin ( x )}

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

:اضرب كسور

= \frac{1 \cdot 5}{sin ( x )}

1 \cdot 5 = 5 :

اضرب الأعداد

= \frac{5}{sin ( x )}

= \frac{cos ( x )}{sin ( x )} + \frac{3 sin ( x )}{cos ( x )} - \frac{5}{sin ( x )}

\frac{cos ( x )}{sin ( x )} - \frac{5}{sin ( x )}

وحّد الكسور:

\frac{cos ( x ) - 5}{sin ( x )}

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

فعّل القانون

= \frac{cos ( x ) - 5}{sin ( x )}

= \frac{cos ( x ) - 5}{sin ( x )} + \frac{3 sin ( x )}{cos ( x )}

sin (x), cos (x)

المضاعف المشترك الأصغر لـ:

sin (x) cos (x)

sin (x), cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

cos (x)

= sin (x) cos (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (x) cos (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{cos ( x ) - 5}{sin ( x )} :

multiply the denominator and numerator by

cos (x)

\frac{cos ( x ) - 5}{sin ( x )} = \frac{( cos ( x ) - 5 ) cos ( x )}{sin ( x ) cos ( x )}

For

\frac{sin ( x ) \cdot 3}{cos ( x )} :

multiply the denominator and numerator by

sin (x)

\frac{sin ( x ) \cdot 3}{cos ( x )} = \frac{sin ( x ) \cdot 3 sin ( x )}{cos ( x ) sin ( x )} = \frac{3 sin ^{2} ( x )}{sin ( x ) cos ( x )}

= \frac{( cos ( x ) - 5 ) cos ( x )}{sin ( x ) cos ( x )} + \frac{3 sin ^{2} ( x )}{sin ( x ) cos ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{( cos ( x ) - 5 ) cos ( x ) + 3 sin ^{2} ( x )}{sin ( x ) cos ( x )}

= \frac{cos ( x ) ( cos ( x ) - 5 ) + 3 sin ^{2} ( x )}{sin ( x ) cos ( x )}

\frac{( - 5 + cos ( x ) ) cos ( x ) + 3 sin ^{2} ( x )}{cos ( x ) sin ( x )} = 0

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

(- 5 + cos (x)) cos (x) + 3 sin^{2} (x) = 0

Rewrite using trig identities

(- 5 + cos (x)) cos (x) + 3 sin^{2} (x)

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

:فعّل نطريّة فيتاغوروس

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

= (- 5 + cos (x)) cos (x) + 3 (1 - cos^{2} (x))

(- 5 + cos (x)) cos (x) + 3 (1 - cos^{2} (x))

بسّط:

- 5 cos (x) - 2 cos^{2} (x) + 3

(- 5 + cos (x)) cos (x) + 3 (1 - cos^{2} (x))

= cos (x) (- 5 + cos (x)) + 3 (1 - cos^{2} (x))

cos (x) (- 5 + cos (x))

وسٌع:

- 5 cos (x) + cos^{2} (x)

cos (x) (- 5 + cos (x))

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = cos (x), b = - 5, c = cos (x)

= cos (x) (- 5) + cos (x) cos (x)

فعّل قوانين سالب-موجب

+ (- a) = - a

= - 5 cos (x) + cos (x) cos (x)

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

cos (x) cos (x)

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

:فعّل قانون القوى

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

= cos^{1 + 1} (x)

1 + 1 = 2 :

اجمع الأعداد

= cos^{2} (x)

= - 5 cos (x) + cos^{2} (x)

= - 5 cos (x) + cos^{2} (x) + 3 (1 - cos^{2} (x))

3 (1 - cos^{2} (x))

وسٌع:

3 - 3 cos^{2} (x)

3 (1 - cos^{2} (x))

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = 3, b = 1, c = cos^{2} (x)

= 3 \cdot 1 - 3 cos^{2} (x)

3 \cdot 1 = 3 :

اضرب الأعداد

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

= - 5 cos (x) + cos^{2} (x) + 3 - 3 cos^{2} (x)

- 5 cos (x) + cos^{2} (x) + 3 - 3 cos^{2} (x)

بسّط:

- 5 cos (x) - 2 cos^{2} (x) + 3

- 5 cos (x) + cos^{2} (x) + 3 - 3 cos^{2} (x)

جمّع التعابير المتشابهة

= - 5 cos (x) + cos^{2} (x) - 3 cos^{2} (x) + 3

cos^{2} (x) - 3 cos^{2} (x) = - 2 cos^{2} (x) :

اجمع العناصر المتشابهة

= - 5 cos (x) - 2 cos^{2} (x) + 3

= - 5 cos (x) - 2 cos^{2} (x) + 3

= - 5 cos (x) - 2 cos^{2} (x) + 3

3 - 2 cos^{2} (x) - 5 cos (x) = 0

بالاستعانة بطريقة التعويض

3 - 2 cos^{2} (x) - 5 cos (x) = 0

cos (x) = u :

على افتراض أنّ

3 - 2 u^{2} - 5 u = 0

3 - 2 u^{2} - 5 u = 0 : u = - 3, u = \frac{1}{2}

3 - 2 u^{2} - 5 u = 0

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

اكتب بالصورة الاعتياديّة

- 2 u^{2} - 5 u + 3 = 0

حلّ بالاستعانة بالصيغة التربيعيّة

- 2 u^{2} - 5 u + 3 = 0

الصيغة لحلّ المعادلة التربيعيّة

a = - 2, b = - 5, c = 3

لـ

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

(- 5)^{2} - 4 (- 2) \cdot 3 = 7

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

- (- a) = a

فعّل القانون

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

زوجيّ n

إذا تحقّق أنّ

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

:فعّل قانون القوى

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 2 \cdot 3

4 \cdot 2 \cdot 3 = 24 :

اضرب الأعداد

= 5^{2} + 24

5^{2} = 25

= 25 + 24

25 + 24 = 49 :

اجمع الأعداد

= 49

49 = 7^{2} :

حلّل العدد لعوامله أوّليّة

= 7^{2}

n a^{n} = a

:فعْل قانون الجذور

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 5 ) \pm 7}{2 ( - 2 )}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 7}{2 ( - 2 )}, u_{2} = \frac{- ( - 5 ) - 7}{2 ( - 2 )}

u = \frac{- ( - 5 ) + 7}{2 ( - 2 )} : - 3

\frac{- ( - 5 ) + 7}{2 ( - 2 )}

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

:احذف الأقواس

= \frac{5 + 7}{- 2 \cdot 2}

5 + 7 = 12 :

اجمع الأعداد

= \frac{12}{- 2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{12}{- 4}

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

: استخدم ميزات الكسور التالية

= - \frac{12}{4}

\frac{12}{4} = 3 :

اقسم الأعداد

= - 3

u = \frac{- ( - 5 ) - 7}{2 ( - 2 )} : \frac{1}{2}

\frac{- ( - 5 ) - 7}{2 ( - 2 )}

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

:احذف الأقواس

= \frac{5 - 7}{- 2 \cdot 2}

5 - 7 = - 2 :

اطرح 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}

حلول المعادلة التربيعيّة هي

u = - 3, u = \frac{1}{2}

u = cos (x)

استبدل مجددًا

cos (x) = - 3, cos (x) = \frac{1}{2}

cos (x) = - 3, cos (x) = \frac{1}{2}

cos (x) = - 3 :

لا يوجد حلّ

cos (x) = - 3

- 1 \leq cos (x) \leq 1

لا يوجد حلّ

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

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

حلول عامّة لـ

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

وحّد الحلول

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

رسم

أمثلة شائعة

3cos^2(x)=1-5sin(x)

3 cos^{2} (x) = 1 - 5 sin (x)

solvefor x,2sin(3x)-1=0

so l v e f or x, 2 sin (3 x) - 1 = 0

2(cos(t))2-cos(t)-1=0

2 (cos (t)) 2 - cos (t) - 1 = 0

cos(c)=cos(7)cos(60)

cos (c) = cos (7) cos (6 0^{\circ})

cos(xpi)=(-1)^n

cos (x π) = (- 1)^{n}