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شائع علم المثلثات >

csc(x)-sin(x)=cos(x)cot(3x-50)

الحلّ

csc (x) - sin (x) = cos (x) cot (3 x - 50)

الحلّ

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = πn + 25, x = \frac{π}{2} + 25 + πn

درجات

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 1432.39448 \dots^{\circ} + 18 0^{\circ} n, x = 1522.39448 \dots^{\circ} + 18 0^{\circ} n

خطوات الحلّ

csc (x) - sin (x) = cos (x) cot (3 x - 50)

من الطرفين

cos (x) cot (3 x - 50)

اطرح

csc (x) - sin (x) - cos (x) cot (3 x - 50) = 0

sin, cos :

عبّر بواسطة

csc (x) - sin (x) - cos (x) cot (- 50 + 3 x)

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

:Use the basic trigonometric identity

= \frac{1}{sin ( x )} - sin (x) - cos (x) cot (- 50 + 3 x)

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

:Use the basic trigonometric identity

= \frac{1}{sin ( x )} - sin (x) - cos (x) \frac{cos ( - 50 + 3 x )}{sin ( - 50 + 3 x )}

\frac{1}{sin ( x )} - sin (x) - cos (x) \frac{cos ( - 50 + 3 x )}{sin ( - 50 + 3 x )}

بسّط:

\frac{sin ( 3 x - 50 ) - sin ^{2} ( x ) sin ( 3 x - 50 ) - cos ( - 50 + 3 x ) cos ( x ) sin ( x )}{sin ( x ) sin ( 3 x - 50 )}

\frac{1}{sin ( x )} - sin (x) - cos (x) \frac{cos ( - 50 + 3 x )}{sin ( - 50 + 3 x )}

cos (x) \frac{cos ( - 50 + 3 x )}{sin ( - 50 + 3 x )}

اضرب بـ:

\frac{cos ( 3 x - 50 ) cos ( x )}{sin ( - 50 + 3 x )}

cos (x) \frac{cos ( - 50 + 3 x )}{sin ( - 50 + 3 x )}

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

:اضرب كسور

= \frac{cos ( - 50 + 3 x ) cos ( x )}{sin ( - 50 + 3 x )}

= \frac{1}{sin ( x )} - sin (x) - \frac{cos ( 3 x - 50 ) cos ( x )}{sin ( 3 x - 50 )}

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

:حوّل الأعداد لكسور

= \frac{1}{sin ( x )} - \frac{sin ( x )}{1} - \frac{cos ( - 50 + 3 x ) cos ( x )}{sin ( - 50 + 3 x )}

sin (x), 1, sin (- 50 + 3 x)

المضاعف المشترك الأصغر لـ:

sin (x) sin (3 x - 50)

sin (x), 1, sin (- 50 + 3 x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear in at least one of the factored expressions

= sin (x) sin (3 x - 50)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (x) sin (3 x - 50)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{1}{sin ( x )} :

multiply the denominator and numerator by

sin (3 x - 50)

\frac{1}{sin ( x )} = \frac{1 \cdot sin ( 3 x - 50 )}{sin ( x ) sin ( 3 x - 50 )} = \frac{sin ( 3 x - 50 )}{sin ( x ) sin ( 3 x - 50 )}

For

\frac{sin ( x )}{1} :

multiply the denominator and numerator by

sin (x) sin (3 x - 50)

\frac{sin ( x )}{1} = \frac{sin ( x ) sin ( x ) sin ( 3 x - 50 )}{1 \cdot sin ( x ) sin ( 3 x - 50 )} = \frac{sin ^{2} ( x ) sin ( 3 x - 50 )}{sin ( x ) sin ( 3 x - 50 )}

For

\frac{cos ( - 50 + 3 x ) cos ( x )}{sin ( - 50 + 3 x )} :

multiply the denominator and numerator by

sin (x)

\frac{cos ( - 50 + 3 x ) cos ( x )}{sin ( - 50 + 3 x )} = \frac{cos ( - 50 + 3 x ) cos ( x ) sin ( x )}{sin ( - 50 + 3 x ) sin ( x )}

= \frac{sin ( 3 x - 50 )}{sin ( x ) sin ( 3 x - 50 )} - \frac{sin ^{2} ( x ) sin ( 3 x - 50 )}{sin ( x ) sin ( 3 x - 50 )} - \frac{cos ( - 50 + 3 x ) cos ( x ) sin ( x )}{sin ( - 50 + 3 x ) sin ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{sin ( 3 x - 50 ) - sin ^{2} ( x ) sin ( 3 x - 50 ) - cos ( - 50 + 3 x ) cos ( x ) sin ( x )}{sin ( x ) sin ( 3 x - 50 )}

= \frac{sin ( 3 x - 50 ) - sin ^{2} ( x ) sin ( 3 x - 50 ) - cos ( - 50 + 3 x ) cos ( x ) sin ( x )}{sin ( x ) sin ( 3 x - 50 )}

\frac{sin ( - 50 + 3 x ) - sin ( - 50 + 3 x ) sin ^{2} ( x ) - cos ( - 50 + 3 x ) cos ( x ) sin ( x )}{sin ( - 50 + 3 x ) sin ( x )} = 0

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

sin (- 50 + 3 x) - sin (- 50 + 3 x) sin^{2} (x) - cos (- 50 + 3 x) cos (x) sin (x) = 0

Rewrite using trig identities

sin (- 50 + 3 x) - sin (- 50 + 3 x) sin^{2} (x) - cos (- 50 + 3 x) cos (x) sin (x)

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

:فعّل نطريّة فيتاغوروس

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

= sin (- 50 + 3 x) - sin (- 50 + 3 x) (1 - cos^{2} (x)) - cos (- 50 + 3 x) cos (x) sin (x)

sin (- 50 + 3 x) - sin (- 50 + 3 x) (1 - cos^{2} (x)) - cos (- 50 + 3 x) cos (x) sin (x)

بسّط:

cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x)

sin (- 50 + 3 x) - sin (- 50 + 3 x) (1 - cos^{2} (x)) - cos (- 50 + 3 x) cos (x) sin (x)

- sin (- 50 + 3 x) (1 - cos^{2} (x))

وسٌع:

- sin (- 50 + 3 x) + cos^{2} (x) sin (- 50 + 3 x)

- sin (- 50 + 3 x) (1 - cos^{2} (x))

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = - sin (- 50 + 3 x), b = 1, c = cos^{2} (x)

= - sin (- 50 + 3 x) \cdot 1 - (- sin (- 50 + 3 x)) cos^{2} (x)

فعّل قوانين سالب-موجب

- (- a) = a

= - 1 \cdot sin (- 50 + 3 x) + cos^{2} (x) sin (- 50 + 3 x)

1 \cdot sin (- 50 + 3 x) = sin (- 50 + 3 x) :

اضرب

= - sin (- 50 + 3 x) + cos^{2} (x) sin (- 50 + 3 x)

= sin (- 50 + 3 x) - sin (- 50 + 3 x) + cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x)

sin (3 x - 50) - sin (3 x - 50) = 0 :

اجمع العناصر المتشابهة

= cos^{2} (x) sin (3 x - 50) - cos (3 x - 50) cos (x) sin (x)

= cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x)

cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x) = 0

cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x)

حلل إلى عوامل:

cos (x) (cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x))

cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x)

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

:فعّل قانون القوى

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

= cos (x) cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x)

cos (x)

قم باخراج العامل المشترك

= cos (x) (cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x))

cos (x) (cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x)) = 0

حلّ كل جزء على حدة

cos (x) = 0 or cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

cos (x) = 0 :

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}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x) = 0 : x = πn + 25, x = \frac{π}{2} + 25 + πn

cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x) = 0

Rewrite using trig identities

cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x)

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

:فعّل متطابقة الطرح لزوايا

= sin (- 50 + 3 x - x)

sin (- 50 + 3 x - x) = 0

sin (- 50 + 3 x - x) = 0 :

حلول عامّة لـ

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

- 50 + 3 x - x = 0 + 2 πn, - 50 + 3 x - x = π + 2 πn

- 50 + 3 x - x = 0 + 2 πn, - 50 + 3 x - x = π + 2 πn

- 50 + 3 x - x = 0 + 2 πn

حلّ:

x = πn + 25

- 50 + 3 x - x = 0 + 2 πn

3 x - x = 2 x :

اجمع العناصر المتشابهة

- 50 + 2 x = 0 + 2 πn

0 + 2 πn = 2 πn

- 50 + 2 x = 2 πn

انقل

50

إلى الجانب الأيمن

- 50 + 2 x = 2 πn

للطرفين

50

أضف

- 50 + 2 x + 50 = 2 πn + 50

بسّط

2 x = 2 πn + 50

2 x = 2 πn + 50

2

اقسم الطرفين على

2 x = 2 πn + 50

2

اقسم الطرفين على

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{50}{2}

بسّط

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{50}{2}

\frac{2 x}{2}

بسّط:

x

\frac{2 x}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= x

\frac{2 πn}{2} + \frac{50}{2}

بسّط:

πn + 25

\frac{2 πn}{2} + \frac{50}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= πn + \frac{50}{2}

\frac{50}{2} = 25 :

اقسم الأعداد

= πn + 25

x = πn + 25

x = πn + 25

x = πn + 25

- 50 + 3 x - x = π + 2 πn

حلّ:

x = \frac{π}{2} + 25 + πn

- 50 + 3 x - x = π + 2 πn

3 x - x = 2 x :

اجمع العناصر المتشابهة

- 50 + 2 x = π + 2 πn

انقل

50

إلى الجانب الأيمن

- 50 + 2 x = π + 2 πn

للطرفين

50

أضف

- 50 + 2 x + 50 = π + 2 πn + 50

بسّط

2 x = π + 2 πn + 50

2 x = π + 2 πn + 50

2

اقسم الطرفين على

2 x = π + 2 πn + 50

2

اقسم الطرفين على

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2} + \frac{50}{2}

بسّط

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2} + \frac{50}{2}

\frac{2 x}{2}

بسّط:

x

\frac{2 x}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= x

\frac{π}{2} + \frac{2 πn}{2} + \frac{50}{2}

بسّط:

\frac{π}{2} + 25 + πn

\frac{π}{2} + \frac{2 πn}{2} + \frac{50}{2}

جمّع التعابير المتشابهة

= \frac{π}{2} + \frac{50}{2} + \frac{2 πn}{2}

\frac{50}{2} = 25 :

اقسم الأعداد

= \frac{π}{2} + 25 + \frac{2 πn}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= \frac{π}{2} + 25 + πn

x = \frac{π}{2} + 25 + πn

x = \frac{π}{2} + 25 + πn

x = \frac{π}{2} + 25 + πn

x = πn + 25, x = \frac{π}{2} + 25 + πn

وحّد الحلول

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = πn + 25, x = \frac{π}{2} + 25 + πn

رسم

أمثلة شائعة

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

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

tan^2(t)-sec^2(t)=1

tan^{2} (t) - sec^{2} (t) = 1

solvefor x,z=sin(2x+3y)

so l v e f or x, z = sin (2 x + 3 y)

4cos(x)+1=2cos(x)

4 cos (x) + 1 = 2 cos (x)

cos(2x)+cos(4x)+cos(6x)=0

cos (2 x) + cos (4 x) + cos (6 x) = 0