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شائع علم المثلثات >

2sec^2(θ)-3=tan^2(θ)

الحلّ

2 sec^{2} (θ) - 3 = tan^{2} (θ)

الحلّ

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

+1

درجات

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

خطوات الحلّ

2 sec^{2} (θ) - 3 = tan^{2} (θ)

من الطرفين

tan^{2} (θ)

اطرح

2 sec^{2} (θ) - 3 - tan^{2} (θ) = 0

Rewrite using trig identities

- 3 - tan^{2} (θ) + 2 sec^{2} (θ)

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

:فعّل نطريّة فيتاغوروس

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

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

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

بسّط:

sec^{2} (θ) - 2

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

- (sec^{2} (θ) - 1) : - sec^{2} (θ) + 1

- (sec^{2} (θ) - 1)

افتح أقواس

= - (sec^{2} (θ)) - (- 1)

فعّل قوانين سالب-موجب

- (- a) = a, - (a) = - a

= - sec^{2} (θ) + 1

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

- 3 - sec^{2} (θ) + 1 + 2 sec^{2} (θ)

بسّط:

sec^{2} (θ) - 2

- 3 - sec^{2} (θ) + 1 + 2 sec^{2} (θ)

جمّع التعابير المتشابهة

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

- sec^{2} (θ) + 2 sec^{2} (θ) = sec^{2} (θ) :

اجمع العناصر المتشابهة

= sec^{2} (θ) - 3 + 1

- 3 + 1 = - 2 :

اطرح/اجمع الأعداد

= sec^{2} (θ) - 2

= sec^{2} (θ) - 2

= sec^{2} (θ) - 2

- 2 + sec^{2} (θ) = 0

بالاستعانة بطريقة التعويض

- 2 + sec^{2} (θ) = 0

sec (θ) = 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

للطرفين

2

أضف

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

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 (θ)

استبدل مجددًا

sec (θ) = 2, sec (θ) = - 2

sec (θ) = 2, sec (θ) = - 2

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

sec (θ) = 2

sec (θ) = 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}

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

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

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

sec (θ) = - 2

sec (θ) = - 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}

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

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

وحّد الحلول

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

رسم

أمثلة شائعة

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

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sin(x)=-sin(2x)

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2cos(2θ)-3cos(θ)-2=0

2 cos (2 θ) - 3 cos (θ) - 2 = 0

tan(2θ)+6tan(θ)+8=0

tan (2 θ) + 6 tan (θ) + 8 = 0