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شائع علم المثلثات >

10sin(x)cos^2(x)+3cos(x)=0

الحلّ

10 sin (x) cos^{2} (x) + 3 cos (x) = 0

الحلّ

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - \frac{0.64350 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.64350 \dots}{2} + πn

درجات

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = - 18.43494 \dots^{\circ} + 18 0^{\circ} n, x = 108.43494 \dots^{\circ} + 18 0^{\circ} n

خطوات الحلّ

10 sin (x) cos^{2} (x) + 3 cos (x) = 0

10 sin (x) cos^{2} (x) + 3 cos (x)

حلل إلى عوامل:

cos (x) (10 sin (x) cos (x) + 3)

10 sin (x) cos^{2} (x) + 3 cos (x)

a^{b + c} = a^{b} a^{c}

:فعّل قانون القوى

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

= 10 cos (x) cos (x) + 3 cos (x)

cos (x)

قم باخراج العامل المشترك

= cos (x) (10 sin (x) cos (x) + 3)

cos (x) (10 sin (x) cos (x) + 3) = 0

حلّ كل جزء على حدة

cos (x) = 0 or 10 sin (x) cos (x) + 3 = 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

10 sin (x) cos (x) + 3 = 0 : x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

10 sin (x) cos (x) + 3 = 0

Rewrite using trig identities

10 sin (x) cos (x) + 3

2 sin (x) cos (x) = sin (2 x)

:فعّل متطابقة الزاوية المضاعفة

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= 3 + 10 \cdot \frac{sin ( 2 x )}{2}

3 + 10 \cdot \frac{sin ( 2 x )}{2} = 0

10 \cdot \frac{sin ( 2 x )}{2} = 5 sin (2 x)

10 \cdot \frac{sin ( 2 x )}{2}

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{sin ( 2 x ) \cdot 10}{2}

\frac{10}{2} = 5 :

اقسم الأعداد

= 5 sin (2 x)

3 + 5 sin (2 x) = 0

انقل

3

إلى الجانب الأيمن

3 + 5 sin (2 x) = 0

من الطرفين

3

اطرح

3 + 5 sin (2 x) - 3 = 0 - 3

بسّط

5 sin (2 x) = - 3

5 sin (2 x) = - 3

5

اقسم الطرفين على

5 sin (2 x) = - 3

5

اقسم الطرفين على

\frac{5 sin ( 2 x )}{5} = \frac{- 3}{5}

بسّط

sin (2 x) = - \frac{3}{5}

sin (2 x) = - \frac{3}{5}

Apply trig inverse properties

sin (2 x) = - \frac{3}{5}

sin (2 x) = - \frac{3}{5} :

حلول عامّة لـ

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

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

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

2 x = arcsin (- \frac{3}{5}) + 2 πn

حلّ:

x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn

2 x = arcsin (- \frac{3}{5}) + 2 πn

arcsin (- \frac{3}{5}) + 2 πn

بسّط:

- arcsin (\frac{3}{5}) + 2 πn

arcsin (- \frac{3}{5}) + 2 πn

arcsin (- x) = - arcsin (x) :

استخدم القانون التالي

arcsin (- \frac{3}{5}) = - arcsin (\frac{3}{5})

= - arcsin (\frac{3}{5}) + 2 πn

2 x = - arcsin (\frac{3}{5}) + 2 πn

2

اقسم الطرفين على

2 x = - arcsin (\frac{3}{5}) + 2 πn

2

اقسم الطرفين على

\frac{2 x}{2} = - \frac{arcsin ( \frac{3}{5} )}{2} + \frac{2 πn}{2}

بسّط

x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn

x = - \frac{arcsin ( \frac{3}{5} )}{2} + πn

2 x = π + arcsin (\frac{3}{5}) + 2 πn

حلّ:

x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

2 x = π + arcsin (\frac{3}{5}) + 2 πn

2

اقسم الطرفين على

2 x = π + arcsin (\frac{3}{5}) + 2 πn

2

اقسم الطرفين على

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + \frac{2 πn}{2}

بسّط

x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

x = \frac{π}{2} + \frac{arcsin ( \frac{3}{5} )}{2} + πn

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

وحّد الحلول

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

أظهر الحلّ بالتمثيل العشريّ

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - \frac{0.64350 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.64350 \dots}{2} + πn

رسم

أمثلة شائعة

(2cos(x))-1=0

(2 cos (x)) - 1 = 0

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

3 cos (x) = sin^{2} (x)

sin(θ)=(5pi)/3

sin (θ) = \frac{5 π}{3}

10cos(θ)=8

10 cos (θ) = 8

cos(θ)=(-18)/(sqrt(20)*\sqrt{26)}

cos (θ) = \frac{- 18}{20 \cdot 26}