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شائع علم المثلثات >

10sin(20t-pi/6)=8

الحلّ

10 sin (20 t - \frac{π}{6}) = 8

الحلّ

t = \frac{πn}{10} + \frac{π}{120} + \frac{0.92729 \dots}{20}, t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{0.92729 \dots}{20}

+1

درجات

t = 4.15650 \dots^{\circ} + 1 8^{\circ} n, t = 7.84349 \dots^{\circ} + 1 8^{\circ} n

خطوات الحلّ

10 sin (20 t - \frac{π}{6}) = 8

10

اقسم الطرفين على

10 sin (20 t - \frac{π}{6}) = 8

10

اقسم الطرفين على

\frac{10 sin ( 20 t - \frac{π}{6} )}{10} = \frac{8}{10}

بسّط

sin (20 t - \frac{π}{6}) = \frac{4}{5}

sin (20 t - \frac{π}{6}) = \frac{4}{5}

Apply trig inverse properties

sin (20 t - \frac{π}{6}) = \frac{4}{5}

sin (20 t - \frac{π}{6}) = \frac{4}{5} :

حلول عامّة لـ

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

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn, 20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn, 20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn

حلّ:

t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn

انقل

\frac{π}{6}

إلى الجانب الأيمن

20 t - \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn

للطرفين

\frac{π}{6}

أضف

20 t - \frac{π}{6} + \frac{π}{6} = arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

بسّط

20 t = arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

20 t = arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

20

اقسم الطرفين على

20 t = arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

20

اقسم الطرفين على

\frac{20 t}{20} = \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

بسّط

\frac{20 t}{20} = \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

\frac{20 t}{20}

بسّط:

t

\frac{20 t}{20}

\frac{20}{20} = 1 :

اقسم الأعداد

= t

\frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

بسّط:

\frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

\frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

جمّع التعابير المتشابهة

= \frac{2 πn}{20} + \frac{\frac{π}{6}}{20} + \frac{arcsin ( \frac{4}{5} )}{20}

\frac{2 πn}{20} = \frac{πn}{10}

\frac{2 πn}{20}

2 :

إلغ العوامل المشتركة

= \frac{πn}{10}

\frac{\frac{π}{6}}{20} = \frac{π}{120}

\frac{\frac{π}{6}}{20}

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

: استخدم ميزات الكسور التالية

= \frac{π}{6 \cdot 20}

6 \cdot 20 = 120 :

اضرب الأعداد

= \frac{π}{120}

= \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}

20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn

حلّ:

t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn

انقل

\frac{π}{6}

إلى الجانب الأيمن

20 t - \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn

للطرفين

\frac{π}{6}

أضف

20 t - \frac{π}{6} + \frac{π}{6} = π - arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

بسّط

20 t = π - arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

20 t = π - arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

20

اقسم الطرفين على

20 t = π - arcsin (\frac{4}{5}) + 2 πn + \frac{π}{6}

20

اقسم الطرفين على

\frac{20 t}{20} = \frac{π}{20} - \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

بسّط

\frac{20 t}{20} = \frac{π}{20} - \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

\frac{20 t}{20}

بسّط:

t

\frac{20 t}{20}

\frac{20}{20} = 1 :

اقسم الأعداد

= t

\frac{π}{20} - \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

بسّط:

\frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

\frac{π}{20} - \frac{arcsin ( \frac{4}{5} )}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20}

جمّع التعابير المتشابهة

= \frac{π}{20} + \frac{2 πn}{20} + \frac{\frac{π}{6}}{20} - \frac{arcsin ( \frac{4}{5} )}{20}

\frac{2 πn}{20} = \frac{πn}{10}

\frac{2 πn}{20}

2 :

إلغ العوامل المشتركة

= \frac{πn}{10}

\frac{\frac{π}{6}}{20} = \frac{π}{120}

\frac{\frac{π}{6}}{20}

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

: استخدم ميزات الكسور التالية

= \frac{π}{6 \cdot 20}

6 \cdot 20 = 120 :

اضرب الأعداد

= \frac{π}{120}

= \frac{π}{20} + \frac{πn}{10} + \frac{π}{120} - \frac{arcsin ( \frac{4}{5} )}{20}

جمّع التعابير المتشابهة

= \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

t = \frac{πn}{10} + \frac{π}{120} + \frac{arcsin ( \frac{4}{5} )}{20}, t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{arcsin ( \frac{4}{5} )}{20}

أظهر الحلّ بالتمثيل العشريّ

t = \frac{πn}{10} + \frac{π}{120} + \frac{0.92729 \dots}{20}, t = \frac{π}{20} + \frac{π}{120} + \frac{πn}{10} - \frac{0.92729 \dots}{20}

رسم

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