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شائع علم المثلثات >

tan(x)-csc(2x)=0

الحلّ

tan (x) - csc (2 x) = 0

الحلّ

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

درجات

x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

خطوات الحلّ

tan (x) - csc (2 x) = 0

sin, cos :

عبّر بواسطة

- csc (2 x) + tan (x)

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

:Use the basic trigonometric identity

= - \frac{1}{sin ( 2 x )} + tan (x)

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

:Use the basic trigonometric identity

= - \frac{1}{sin ( 2 x )} + \frac{sin ( x )}{cos ( x )}

- \frac{1}{sin ( 2 x )} + \frac{sin ( x )}{cos ( x )}

بسّط:

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

- \frac{1}{sin ( 2 x )} + \frac{sin ( x )}{cos ( x )}

sin (2 x), cos (x)

المضاعف المشترك الأصغر لـ:

sin (2 x) cos (x)

sin (2 x), cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (2 x)

cos (x)

= sin (2 x) cos (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

sin (2 x) cos (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

\frac{1}{sin ( 2 x )} :

multiply the denominator and numerator by

cos (x)

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

For

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

multiply the denominator and numerator by

sin (2 x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

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

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

Rewrite using trig identities

- cos (x) + sin (2 x) sin (x)

sin (2 x) = 2 sin (x) cos (x)

:فعّل متطابقة الزاوية المضاعفة

= - cos (x) + 2 sin (x) cos (x) sin (x)

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

2 sin (x) cos (x) sin (x)

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

:فعّل قانون القوى

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

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

1 + 1 = 2 :

اجمع الأعداد

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

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

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

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

حلل إلى عوامل:

cos (x) (2 sin (x) + 1) (2 sin (x) - 1)

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

cos (x)

قم باخراج العامل المشترك

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

2 sin^{2} (x) - 1

حلل إلى عوامل:

(2 sin (x) + 1) (2 sin (x) - 1)

2 sin^{2} (x) - 1

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

كـ

2 sin^{2} (x) - 1

اكتب مجددًا

2 sin^{2} (x) - 1

a = (a)^{2}

:فعْل قانون الجذور

2 = (2)^{2}

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

1^{2}

كـ

1

اكتب مجددًا

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

a^{m} b^{m} = (ab)^{m}

:فعّل قانون القوى

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

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

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

x^{2} - y^{2} = (x + y) (x - y)

فعّل قانون فرق المربّعات

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

= (2 sin (x) + 1) (2 sin (x) - 1)

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

cos (x) (2 sin (x) + 1) (2 sin (x) - 1) = 0

حلّ كل جزء على حدة

cos (x) = 0 or 2 sin (x) + 1 = 0 or 2 sin (x) - 1 = 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

2 sin (x) + 1 = 0 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

2 sin (x) + 1 = 0

انقل

1

إلى الجانب الأيمن

2 sin (x) + 1 = 0

من الطرفين

1

اطرح

2 sin (x) + 1 - 1 = 0 - 1

بسّط

2 sin (x) = - 1

2 sin (x) = - 1

2

اقسم الطرفين على

2 sin (x) = - 1

2

اقسم الطرفين على

\frac{2 sin ( x )}{2} = \frac{- 1}{2}

بسّط

\frac{2 sin ( x )}{2} = \frac{- 1}{2}

\frac{2 sin ( x )}{2}

بسّط:

sin (x)

\frac{2 sin ( x )}{2}

2 :

إلغ العوامل المشتركة

= sin (x)

\frac{- 1}{2}

بسّط:

- \frac{2}{2}

\frac{- 1}{2}

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

: استخدم ميزات الكسور التالية

= - \frac{1}{2}

- \frac{1}{2}

حوّل لصيغة عدد كسريّ:

- \frac{2}{2}

- \frac{1}{2}

\frac{2}{2}

اضرب بالمرافق

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

a a = a

:فعْل قانون الجذور

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

sin (x) = - \frac{2}{2}

sin (x) = - \frac{2}{2}

sin (x) = - \frac{2}{2}

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

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}

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

2 sin (x) - 1 = 0 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

2 sin (x) - 1 = 0

انقل

1

إلى الجانب الأيمن

2 sin (x) - 1 = 0

للطرفين

1

أضف

2 sin (x) - 1 + 1 = 0 + 1

بسّط

2 sin (x) = 1

2 sin (x) = 1

2

اقسم الطرفين على

2 sin (x) = 1

2

اقسم الطرفين على

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

بسّط

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

\frac{2 sin ( x )}{2}

بسّط:

sin (x)

\frac{2 sin ( x )}{2}

2 :

إلغ العوامل المشتركة

= sin (x)

\frac{1}{2}

بسّط:

\frac{2}{2}

\frac{1}{2}

\frac{2}{2}

اضرب بالمرافق

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

a a = a

:فعْل قانون الجذور

22 = 2

= 2

= \frac{2}{2}

sin (x) = \frac{2}{2}

sin (x) = \frac{2}{2}

sin (x) = \frac{2}{2}

sin (x) = \frac{2}{2} :

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}

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

وحّد الحلول

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn :

بما أنّ المعادلة غير معرّفة لـ

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

رسم

أمثلة شائعة

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