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شائع علم المثلثات >

sin(pi/2 (x+1))= 2/5

الحلّ

sin (\frac{π}{2} (x + 1)) = \frac{2}{5}

الحلّ

x = \frac{2 \cdot 0.41151 \dots}{π} + 4 n - 1, x = 4 n + 1 - \frac{2 \cdot 0.41151 \dots}{π}

+1

درجات

x = - 42.28544 \dots^{\circ} + 229.18311 \dots^{\circ} n, x = 42.28544 \dots^{\circ} + 229.18311 \dots^{\circ} n

خطوات الحلّ

sin (\frac{π}{2} (x + 1)) = \frac{2}{5}

Apply trig inverse properties

sin (\frac{π}{2} (x + 1)) = \frac{2}{5}

sin (\frac{π}{2} (x + 1)) = \frac{2}{5} :

حلول عامّة لـ

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

\frac{π}{2} (x + 1) = arcsin (\frac{2}{5}) + 2 πn, \frac{π}{2} (x + 1) = π - arcsin (\frac{2}{5}) + 2 πn

\frac{π}{2} (x + 1) = arcsin (\frac{2}{5}) + 2 πn, \frac{π}{2} (x + 1) = π - arcsin (\frac{2}{5}) + 2 πn

\frac{π}{2} (x + 1) = arcsin (\frac{2}{5}) + 2 πn

حلّ:

x = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

\frac{π}{2} (x + 1) = arcsin (\frac{2}{5}) + 2 πn

2

اضرب الطرفين بـ

\frac{π}{2} (x + 1) = arcsin (\frac{2}{5}) + 2 πn

2

اضرب الطرفين بـ

2 \cdot \frac{π}{2} (x + 1) = 2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

بسّط

2 \cdot \frac{π}{2} (x + 1) = 2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

2 \cdot \frac{π}{2} (x + 1)

بسّط:

π (x + 1)

2 \cdot \frac{π}{2} (x + 1)

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

:اضرب كسور

= \frac{2 π}{2} (x + 1)

2 :

إلغ العوامل المشتركة

= (x + 1) π

2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

بسّط:

2 arcsin (\frac{2}{5}) + 4 πn

2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

2 \cdot 2 = 4 :

اضرب الأعداد

= 2 arcsin (\frac{2}{5}) + 4 πn

π (x + 1) = 2 arcsin (\frac{2}{5}) + 4 πn

π (x + 1) = 2 arcsin (\frac{2}{5}) + 4 πn

π (x + 1) = 2 arcsin (\frac{2}{5}) + 4 πn

π

اقسم الطرفين على

π (x + 1) = 2 arcsin (\frac{2}{5}) + 4 πn

π

اقسم الطرفين على

\frac{π ( x + 1 )}{π} = \frac{2 arcsin ( \frac{2}{5} )}{π} + \frac{4 πn}{π}

بسّط

x + 1 = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

x + 1 = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

انقل

1

إلى الجانب الأيمن

x + 1 = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

من الطرفين

1

اطرح

x + 1 - 1 = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

بسّط

x = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

x = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

\frac{π}{2} (x + 1) = π - arcsin (\frac{2}{5}) + 2 πn

حلّ:

x = 4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

\frac{π}{2} (x + 1) = π - arcsin (\frac{2}{5}) + 2 πn

2

اضرب الطرفين بـ

\frac{π}{2} (x + 1) = π - arcsin (\frac{2}{5}) + 2 πn

2

اضرب الطرفين بـ

2 \cdot \frac{π}{2} (x + 1) = 2 π - 2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

بسّط

2 \cdot \frac{π}{2} (x + 1) = 2 π - 2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

2 \cdot \frac{π}{2} (x + 1)

بسّط:

π (x + 1)

2 \cdot \frac{π}{2} (x + 1)

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

:اضرب كسور

= \frac{2 π}{2} (x + 1)

2 :

إلغ العوامل المشتركة

= (x + 1) π

2 π - 2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

بسّط:

2 π - 2 arcsin (\frac{2}{5}) + 4 πn

2 π - 2 arcsin (\frac{2}{5}) + 2 \cdot 2 πn

2 \cdot 2 = 4 :

اضرب الأعداد

= 2 π - 2 arcsin (\frac{2}{5}) + 4 πn

π (x + 1) = 2 π - 2 arcsin (\frac{2}{5}) + 4 πn

π (x + 1) = 2 π - 2 arcsin (\frac{2}{5}) + 4 πn

π (x + 1) = 2 π - 2 arcsin (\frac{2}{5}) + 4 πn

π

اقسم الطرفين على

π (x + 1) = 2 π - 2 arcsin (\frac{2}{5}) + 4 πn

π

اقسم الطرفين على

\frac{π ( x + 1 )}{π} = \frac{2 π}{π} - \frac{2 arcsin ( \frac{2}{5} )}{π} + \frac{4 πn}{π}

بسّط

\frac{π ( x + 1 )}{π} = \frac{2 π}{π} - \frac{2 arcsin ( \frac{2}{5} )}{π} + \frac{4 πn}{π}

\frac{π ( x + 1 )}{π}

بسّط:

x + 1

\frac{π ( x + 1 )}{π}

π :

إلغ العوامل المشتركة

= x + 1

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

بسّط:

2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

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

\frac{2 π}{π}

اختزل:

2

\frac{2 π}{π}

π :

إلغ العوامل المشتركة

= 2

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

\frac{4 πn}{π}

اختزل:

4 n

\frac{4 πn}{π}

π :

إلغ العوامل المشتركة

= 4 n

= 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

x + 1 = 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

x + 1 = 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

x + 1 = 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

انقل

1

إلى الجانب الأيمن

x + 1 = 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n

من الطرفين

1

اطرح

x + 1 - 1 = 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

بسّط

x + 1 - 1 = 2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

x + 1 - 1

بسّط:

x

x + 1 - 1

1 - 1 = 0 :

اجمع العناصر المتشابهة

= x

2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

بسّط:

4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

2 - \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1

2 - 1 = 1 :

اطرح الأعداد

= 4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

x = 4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

x = 4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

x = 4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

x = \frac{2 arcsin ( \frac{2}{5} )}{π} + 4 n - 1, x = 4 n + 1 - \frac{2 arcsin ( \frac{2}{5} )}{π}

أظهر الحلّ بالتمثيل العشريّ

x = \frac{2 \cdot 0.41151 \dots}{π} + 4 n - 1, x = 4 n + 1 - \frac{2 \cdot 0.41151 \dots}{π}

رسم

أمثلة شائعة

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(2sin^2(x)-1)(tan(x)-1)=0

(2 sin^{2} (x) - 1) (tan (x) - 1) = 0

(csc(x)+2)(csc(x)-sqrt(2))=0,0<= x<= 2pi

(csc (x) + 2) (csc (x) - 2) = 0, 0 \leq x \leq 2 π

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