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2sec^2(θ)+3=4+sec^2(θ)

الحلّ

2 sec^{2} (θ) + 3 = 4 + sec^{2} (θ)

الحلّ

θ = 2 πn, θ = π + 2 πn

+1

درجات

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

2 sec^{2} (θ) + 3 = 4 + sec^{2} (θ)

بالاستعانة بطريقة التعويض

2 sec^{2} (θ) + 3 = 4 + sec^{2} (θ)

sec (θ) = u :

على افتراض أنّ

2 u^{2} + 3 = 4 + u^{2}

2 u^{2} + 3 = 4 + u^{2} : u = 1, u = - 1

2 u^{2} + 3 = 4 + u^{2}

انقل

3

إلى الجانب الأيمن

2 u^{2} + 3 = 4 + u^{2}

من الطرفين

3

اطرح

2 u^{2} + 3 - 3 = 4 + u^{2} - 3

بسّط

2 u^{2} = u^{2} + 1

2 u^{2} = u^{2} + 1

انقل

u^{2}

إلى الجانب الأيسر

2 u^{2} = u^{2} + 1

من الطرفين

u^{2}

اطرح

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

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 = sec (θ)

استبدل مجددًا

sec (θ) = 1, sec (θ) = - 1

sec (θ) = 1, sec (θ) = - 1

sec (θ) = 1 : θ = 2 πn

sec (θ) = 1

sec (θ) = 1 :

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}

θ = 0 + 2 πn

θ = 0 + 2 πn

θ = 0 + 2 πn

حلّ:

θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn

sec (θ) = - 1 : θ = π + 2 πn

sec (θ) = - 1

sec (θ) = - 1 :

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}

θ = π + 2 πn

θ = π + 2 πn

وحّد الحلول

θ = 2 πn, θ = π + 2 πn

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