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شائع علم المثلثات >

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

الحلّ

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

الحلّ

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

+1

درجات

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

خطوات الحلّ

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

من الطرفين

2 sec^{2} (x) - 3

اطرح

tan^{2} (x) - 2 sec^{2} (x) + 3 = 0

Rewrite using trig identities

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

tan^{2} (x) + 1 = sec^{2} (x)

:فعّل نطريّة فيتاغوروس

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

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

3 + sec^{2} (x) - 1 - 2 sec^{2} (x)

بسّط:

- sec^{2} (x) + 2

3 + sec^{2} (x) - 1 - 2 sec^{2} (x)

جمّع التعابير المتشابهة

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

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

اجمع العناصر المتشابهة

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

3 - 1 = 2 :

اطرح/اجمع الأعداد

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

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

2 - sec^{2} (x) = 0

بالاستعانة بطريقة التعويض

2 - sec^{2} (x) = 0

sec (x) = u :

على افتراض أنّ

2 - u^{2} = 0

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

2 - u^{2} = 0

انقل

2

إلى الجانب الأيمن

2 - u^{2} = 0

من الطرفين

2

اطرح

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

بسّط

- u^{2} = - 2

- u^{2} = - 2

- 1

اقسم الطرفين على

- u^{2} = - 2

- 1

اقسم الطرفين على

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

بسّط

u^{2} = 2

u^{2} = 2

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

الحلول هي

x^{2} = f (a)

لـ

u = 2, u = - 2

u = sec (x)

استبدل مجددًا

sec (x) = 2, sec (x) = - 2

sec (x) = 2, sec (x) = - 2

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

sec (x) = 2

sec (x) = 2 :

حلول عامّة لـ

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

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

sec (x) = - 2 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sec (x) = - 2

sec (x) = - 2 :

حلول عامّة لـ

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

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

وحّد الحلول

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

رسم

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