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شائع علم المثلثات >

(tan(x))(sin^2(x)-a)(sec(x)-a)=0

الحلّ

(tan (x)) (sin^{2} (x) - a) (sec (x) - a) = 0

الحلّ

x = πn, x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn, x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn, x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

خطوات الحلّ

(tan (x)) (sin^{2} (x) - a) (sec (x) - a) = 0

حلّ كل جزء على حدة

tan (x) = 0 or sin^{2} (x) - a = 0 or sec (x) - a = 0

tan (x) = 0 : x = πn

tan (x) = 0

tan (x) = 0 :

حلول عامّة لـ

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

x = 0 + πn

حلّ:

x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

sin^{2} (x) - a = 0 : x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn, x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

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

بالاستعانة بطريقة التعويض

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

sin (x) = u :

على افتراض أنّ

u^{2} - a = 0

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

u^{2} - a = 0

انقل

a

إلى الجانب الأيمن

u^{2} - a = 0

للطرفين

a

أضف

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

بسّط

u^{2} = a

u^{2} = a

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

الحلول هي

x^{2} = f (a)

لـ

u = a, u = - a

u = sin (x)

استبدل مجددًا

sin (x) = a, sin (x) = - a

sin (x) = a, sin (x) = - a

sin (x) = a : x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn

sin (x) = a

Apply trig inverse properties

sin (x) = a

sin (x) = a :

حلول عامّة لـ

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn

x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn

sin (x) = - a : x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

sin (x) = - a

Apply trig inverse properties

sin (x) = - a

sin (x) = - a :

حلول عامّة لـ

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

وحّد الحلول

x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn, x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

sec (x) - a = 0 : x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

sec (x) - a = 0

انقل

a

إلى الجانب الأيمن

sec (x) - a = 0

للطرفين

a

أضف

sec (x) - a + a = 0 + a

بسّط

sec (x) = a

sec (x) = a

Apply trig inverse properties

sec (x) = a

sec (x) = a :

حلول عامّة لـ

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

وحّد الحلول

x = πn, x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn, x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn, x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

رسم

أمثلة شائعة

3cos(x)+2sin(x)=sqrt(3)

3 cos (x) + 2 sin (x) = 3

sin(x)-1=sqrt(3)cos(x)

sin (x) - 1 = 3 cos (x)

cos(3x)=-cos(x)

cos (3 x) = - cos (x)

sqrt(3)sin(x)-((sqrt(3)sin(x))-1)=1n

3 sin (x) - ((3 sin (x)) - 1) = 1 n

3 = 3 - 3 cos (θ)