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شائع علم المثلثات >

0=1.9cos(0.1t)-1.25sin(0.2t)

الحلّ

0 = 1.9 cos (0.1 t) - 1.25 sin (0.2 t)

الحلّ

t = 5 π + 20 πn, t = 15 π + 20 πn, t = \frac{0.86331 \dots + 2 πn}{0.1}, t = \frac{π - 0.86331 \dots + 2 πn}{0.1}

درجات

t = 90 0^{\circ} + 360 0^{\circ} n, t = 270 0^{\circ} + 360 0^{\circ} n, t = 494.64197 \dots^{\circ} + 360 0^{\circ} n, t = 1305.35802 \dots^{\circ} + 360 0^{\circ} n

خطوات الحلّ

0 = 1.9 cos (0.1 t) - 1.25 sin (0.2 t)

بدّل الأطراف

1.9 cos (0.1 t) - 1.25 sin (0.2 t) = 0

u = 0.1 t :

على افتراض أنّ

1.9 cos (u) - 1.25 sin (2 u) = 0

Rewrite using trig identities

- 1.25 sin (2 u) + 1.9 cos (u)

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

:فعّل متطابقة الزاوية المضاعفة

= - 1.25 \cdot 2 sin (u) cos (u) + 1.9 cos (u)

بسّط

= - 2.5 sin (u) cos (u) + 1.9 cos (u)

1.9 cos (u) - 2.5 cos (u) sin (u) = 0

1.9 cos (u) - 2.5 cos (u) sin (u)

حلل إلى عوامل:

- cos (u) (2.5 sin (u) - 1.9)

1.9 cos (u) - 2.5 cos (u) sin (u)

- cos (u)

قم باخراج العامل المشترك

= - cos (u) (- 1.9 + 2.5 sin (u))

- cos (u) (2.5 sin (u) - 1.9) = 0

حلّ كل جزء على حدة

cos (u) = 0 or 2.5 sin (u) - 1.9 = 0

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

cos (u) = 0

cos (u) = 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}

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

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

2.5 sin (u) - 1.9 = 0 : u = arcsin (\frac{19}{25}) + 2 πn, u = π - arcsin (\frac{19}{25}) + 2 πn

2.5 sin (u) - 1.9 = 0

10

اضرب الطرفين بـ

2.5 sin (u) - 1.9 = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

2.5 sin (u) \cdot 10 - 1.9 \cdot 10 = 0 \cdot 10

بسّط

25 sin (u) - 19 = 0

25 sin (u) - 19 = 0

انقل

19

إلى الجانب الأيمن

25 sin (u) - 19 = 0

للطرفين

19

أضف

25 sin (u) - 19 + 19 = 0 + 19

بسّط

25 sin (u) = 19

25 sin (u) = 19

25

اقسم الطرفين على

25 sin (u) = 19

25

اقسم الطرفين على

\frac{25 sin ( u )}{25} = \frac{19}{25}

بسّط

sin (u) = \frac{19}{25}

sin (u) = \frac{19}{25}

Apply trig inverse properties

sin (u) = \frac{19}{25}

sin (u) = \frac{19}{25} :

حلول عامّة لـ

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

u = arcsin (\frac{19}{25}) + 2 πn, u = π - arcsin (\frac{19}{25}) + 2 πn

u = arcsin (\frac{19}{25}) + 2 πn, u = π - arcsin (\frac{19}{25}) + 2 πn

وحّد الحلول

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn, u = arcsin (\frac{19}{25}) + 2 πn, u = π - arcsin (\frac{19}{25}) + 2 πn

u = 0.1 t

استبدل مجددًا

0.1 t = \frac{π}{2} + 2 πn : t = 5 π + 20 πn

0.1 t = \frac{π}{2} + 2 πn

10

اضرب الطرفين بـ

0.1 t = \frac{π}{2} + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.1 t \cdot 10 = \frac{π}{2} \cdot 10 + 2 πn \cdot 10

بسّط

t = 5 π + 20 πn

t = 5 π + 20 πn

0.1 t = \frac{3 π}{2} + 2 πn : t = 15 π + 20 πn

0.1 t = \frac{3 π}{2} + 2 πn

10

اضرب الطرفين بـ

0.1 t = \frac{3 π}{2} + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.1 t \cdot 10 = \frac{3 π}{2} \cdot 10 + 2 πn \cdot 10

بسّط

t = 15 π + 20 πn

t = 15 π + 20 πn

0.1 t = arcsin (\frac{19}{25}) + 2 πn : t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

0.1 t = arcsin (\frac{19}{25}) + 2 πn

0.1

اقسم الطرفين على

0.1 t = arcsin (\frac{19}{25}) + 2 πn

0.1

اقسم الطرفين على

\frac{0.1 t}{0.1} = \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

بسّط

\frac{0.1 t}{0.1} = \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

\frac{0.1 t}{0.1}

بسّط:

t

\frac{0.1 t}{0.1}

0.1 :

إلغ العوامل المشتركة

= t

\frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

بسّط:

\frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

\frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

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

فعّل القانون

= \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

0.1 t = π - arcsin (\frac{19}{25}) + 2 πn : t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

0.1 t = π - arcsin (\frac{19}{25}) + 2 πn

0.1

اقسم الطرفين على

0.1 t = π - arcsin (\frac{19}{25}) + 2 πn

0.1

اقسم الطرفين على

\frac{0.1 t}{0.1} = \frac{π}{0.1} - \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

بسّط

\frac{0.1 t}{0.1} = \frac{π}{0.1} - \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

\frac{0.1 t}{0.1}

بسّط:

t

\frac{0.1 t}{0.1}

0.1 :

إلغ العوامل المشتركة

= t

\frac{π}{0.1} - \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

بسّط:

\frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

\frac{π}{0.1} - \frac{arcsin ( \frac{19}{25} )}{0.1} + \frac{2 πn}{0.1}

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

فعّل القانون

= \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

t = 5 π + 20 πn, t = 15 π + 20 πn, t = \frac{arcsin ( \frac{19}{25} ) + 2 πn}{0.1}, t = \frac{π - arcsin ( \frac{19}{25} ) + 2 πn}{0.1}

أظهر الحلّ بالتمثيل العشريّ

t = 5 π + 20 πn, t = 15 π + 20 πn, t = \frac{0.86331 \dots + 2 πn}{0.1}, t = \frac{π - 0.86331 \dots + 2 πn}{0.1}

رسم

أمثلة شائعة

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sin (\frac{6}{x}) = 36

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tan (x) = \frac{200}{500}