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شائع علم المثلثات >

(-4cos(x)-6sin(x))^2-20sin^2(x)=16

الحلّ

(- 4 cos (x) - 6 sin (x))^{2} - 20 sin^{2} (x) = 16

الحلّ

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

+1

درجات

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

(- 4 cos (x) - 6 sin (x))^{2} - 20 sin^{2} (x) = 16

من الطرفين

16

اطرح

16 cos^{2} (x) + 48 cos (x) sin (x) + 16 sin^{2} (x) - 16 = 0

16 cos^{2} (x) + 48 cos (x) sin (x) + 16 sin^{2} (x) - 16

حلل إلى عوامل:

16 (cos^{2} (x) + 3 sin (x) cos (x) + sin^{2} (x) - 1)

16 cos^{2} (x) + 48 cos (x) sin (x) + 16 sin^{2} (x) - 16

3 \cdot 16

كـ

48

اكتب مجددًا

= 16 cos^{2} (x) + 3 \cdot 16 sin (x) cos (x) + 16 sin^{2} (x) - 16

16

قم باخراج العامل المشترك

= 16 (cos^{2} (x) + 3 sin (x) cos (x) + sin^{2} (x) - 1)

16 (cos^{2} (x) + 3 sin (x) cos (x) + sin^{2} (x) - 1) = 0

Rewrite using trig identities

16 (cos^{2} (x) + 3 sin (x) cos (x) + sin^{2} (x) - 1)

cos^{2} (x) + sin^{2} (x) = 1

:فعّل نطريّة فيتاغوروس

= 16 (- 1 + 3 cos (x) sin (x) + 1)

16 (- 1 + 3 cos (x) sin (x) + 1)

بسّط:

48 cos (x) sin (x)

16 (- 1 + 3 cos (x) sin (x) + 1)

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

بسّط:

3 cos (x) sin (x)

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

جمّع التعابير المتشابهة

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

- 1 + 1 = 0

= 3 cos (x) sin (x)

= 16 \cdot 3 cos (x) sin (x)

16 \cdot 3 = 48 :

اضرب الأعداد

= 48 cos (x) sin (x)

= 48 cos (x) sin (x)

48 cos (x) sin (x) = 0

حلّ كل جزء على حدة

cos (x) = 0 or sin (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

cos (x) = 0 :

حلول عامّة لـ

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0 :

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn

حلّ:

x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

وحّد الحلول

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

رسم

أمثلة شائعة

sin (u) = \frac{5}{13}

0.8 = cos (x)

cos (x) = \frac{12}{20}

2sin^2(x)+sin(x)=1,-pi<= x<= pi

2 sin^{2} (x) + sin (x) = 1, - π \leq x \leq π

cos^2(x)+sin(x)=1,0<= x<= 2pi

cos^{2} (x) + sin (x) = 1, 0 \leq x \leq 2 π