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شائع علم المثلثات >

4sec^2(x)-3tan^2(x)=5tan^2(x)

الحلّ

4 sec^{2} (x) - 3 tan^{2} (x) = 5 tan^{2} (x)

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

4 sec^{2} (x) - 3 tan^{2} (x) = 5 tan^{2} (x)

من الطرفين

5 tan^{2} (x)

اطرح

4 sec^{2} (x) - 8 tan^{2} (x) = 0

4 sec^{2} (x) - 8 tan^{2} (x)

حلل إلى عوامل:

4 (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

4 sec^{2} (x) - 8 tan^{2} (x)

2 \cdot 4

كـ

- 8

اكتب مجددًا

= 4 sec^{2} (x) + 2 \cdot 4 tan^{2} (x)

4

قم باخراج العامل المشترك

= 4 (sec^{2} (x) - 2 tan^{2} (x))

sec^{2} (x) - 2 tan^{2} (x)

حلل إلى عوامل:

(sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

sec^{2} (x) - 2 tan^{2} (x)

sec^{2} (x) - (2 tan (x))^{2}

كـ

sec^{2} (x) - 2 tan^{2} (x)

اكتب مجددًا

sec^{2} (x) - 2 tan^{2} (x)

a = (a)^{2}

:فعْل قانون الجذور

2 = (2)^{2}

= sec^{2} (x) - (2)^{2} tan^{2} (x)

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

:فعّل قانون القوى

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

= sec^{2} (x) - (2 tan (x))^{2}

= sec^{2} (x) - (2 tan (x))^{2}

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

فعّل قانون فرق المربّعات

sec^{2} (x) - (2 tan (x))^{2} = (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

= (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

= 4 (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

4 (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x)) = 0

حلّ كل جزء على حدة

sec (x) + 2 tan (x) = 0 or sec (x) - 2 tan (x) = 0

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

sec (x) + 2 tan (x) = 0

sin, cos :

عبّر بواسطة

sec (x) + 2 tan (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

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

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

فعّل القانون

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

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

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

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

1 + sin (x) 2 = 0

انقل

1

إلى الجانب الأيمن

1 + sin (x) 2 = 0

من الطرفين

1

اطرح

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

بسّط

sin (x) 2 = - 1

sin (x) 2 = - 1

2

اقسم الطرفين على

sin (x) 2 = - 1

2

اقسم الطرفين على

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

بسّط

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

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

بسّط:

sin (x)

\frac{sin ( x ) 2}{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

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

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

sin, cos :

عبّر بواسطة

sec (x) - 2 tan (x)

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

:Use the basic trigonometric identity

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

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

اضرب بـ:

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

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

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

:اضرب كسور

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

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

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

فعّل القانون

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

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

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

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

1 - sin (x) 2 = 0

انقل

1

إلى الجانب الأيمن

1 - sin (x) 2 = 0

من الطرفين

1

اطرح

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

بسّط

- sin (x) 2 = - 1

- sin (x) 2 = - 1

- 2

اقسم الطرفين على

- sin (x) 2 = - 1

- 2

اقسم الطرفين على

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

بسّط

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

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

بسّط:

sin (x)

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

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

: استخدم ميزات الكسور التالية

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

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