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csc(x)sec(x)=tan(x)

الحلّ

csc (x) sec (x) = tan (x)

الحلّ

x \in R لا يوجد حلّ لـ

خطوات الحلّ

csc (x) sec (x) = tan (x)

من الطرفين

tan (x)

اطرح

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

sin, cos :

عبّر بواسطة

- tan (x) + csc (x) sec (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

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

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

:اضرب كسور

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

1 \cdot 1 = 1 :

اضرب الأعداد

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

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

cos (x), sin (x) cos (x)

المضاعف المشترك الأصغر لـ:

cos (x) sin (x)

cos (x), sin (x) cos (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (x)

sin (x) cos (x)

= cos (x) sin (x)

اكتب مجددًا الكسور بحيث يكون المقام مشترك

cos (x) sin (x)

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

sin (x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

\frac{1 - sin ^{2} ( x )}{cos ( x ) sin ( x )} = 0

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

1 - sin^{2} (x) = 0

بالاستعانة بطريقة التعويض

1 - sin^{2} (x) = 0

sin (x) = u :

على افتراض أنّ

1 - u^{2} = 0

1 - u^{2} = 0 : u = 1, u = - 1

1 - u^{2} = 0

انقل

1

إلى الجانب الأيمن

1 - u^{2} = 0

من الطرفين

1

اطرح

1 - u^{2} - 1 = 0 - 1

بسّط

- u^{2} = - 1

- u^{2} = - 1

- 1

اقسم الطرفين على

- u^{2} = - 1

- 1

اقسم الطرفين على

\frac{- u ^{2}}{- 1} = \frac{- 1}{- 1}

بسّط

u^{2} = 1

u^{2} = 1

x = f (a), - f (a)

الحلول هي

x^{2} = f (a)

لـ

u = 1, u = - 1

1 = 1

1

1 = 1

فعّل القانون

= 1

- 1 = - 1

- 1

1 = 1

فعّل القانون

= - 1

u = 1, u = - 1

u = sin (x)

استبدل مجددًا

sin (x) = 1, sin (x) = - 1

sin (x) = 1, sin (x) = - 1

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

sin (x) = 1 :

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}{2} + 2 πn

x = \frac{π}{2} + 2 πn

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

sin (x) = - 1

sin (x) = - 1 :

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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

وحّد الحلول

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

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

بما أنّ المعادلة غير معرّفة لـ

x \in R لا يوجد حلّ لـ

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