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شائع علم المثلثات >

8/150.58tan(45)sqrt(29.81)*0.1^{5/2}

الحلّ

\frac{8}{15} \cdot 0.58 \cdot tan (4 5^{\circ}) \cdot 2 \cdot 9.81 \cdot 0. 1^{\frac{5}{2}}

الحلّ

\frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

عشري

0.00433 \dots

خطوات الحلّ

\frac{8}{15} \cdot 0.58 tan (4 5^{\circ}) 2 \cdot 9.81 \cdot 0. 1^{\frac{5}{2}}

= \frac{8}{15} \cdot \frac{29}{50} tan (4 5^{\circ}) 2 \cdot \frac{981}{100} (\frac{1}{10})^{\frac{5}{2}}

استخدم المتطابقة الأساسيّة التالية:

tan (4 5^{\circ}) = 1

tan (4 5^{\circ})

tan (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 1

= \frac{8}{15} \cdot \frac{29}{50} \cdot 1 \cdot 2 \cdot \frac{981}{100} (\frac{1}{10})^{\frac{5}{2}}

\frac{8}{15} \cdot \frac{29}{50} \cdot 1 \cdot 2 \cdot \frac{981}{100} (\frac{1}{10})^{\frac{5}{2}}

بسّط:

\frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

\frac{8}{15} \cdot \frac{29}{50} \cdot 1 \cdot 2 \cdot \frac{981}{100} (\frac{1}{10})^{\frac{5}{2}}

2 \cdot \frac{981}{100} = \frac{3 \cdot 10 9 ^{\frac{1}{2}}}{5 \cdot 2 ^{\frac{1}{2}}}

2 \cdot \frac{981}{100}

2 \cdot \frac{981}{100}

اضرب بـ:

\frac{981}{50}

2 \cdot \frac{981}{100}

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

:اضرب كسور

= \frac{981 \cdot 2}{100}

981 \cdot 2 = 1962 :

اضرب الأعداد

= \frac{1962}{100}

2 :

إلغ العوامل المشتركة

= \frac{981}{50}

= \frac{981}{50}

a \geq 0, b \geq 0

بافتراض أنّ

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

:فعّل قانون الجذور

= \frac{98 1 ^{\frac{1}{2}}}{5 0 ^{\frac{1}{2}}}

5 0^{\frac{1}{2}} = 5 \cdot 2^{\frac{1}{2}}

5 0^{\frac{1}{2}}

50

تحليل لعوامل أوّليّة لـ:

2 \cdot 5^{2}

50

50 = 25 \cdot 2, 2

ينقسم على

50

= 2 \cdot 25

25 = 5 \cdot 5, 5

ينقسم على

25

= 2 \cdot 5 \cdot 5

مركّب من أعداد أوّليّة فقط، لذلك تحليل إضافيّ غير ممكن

2, 5

= 2 \cdot 5 \cdot 5

= 2 \cdot 5^{2}

= (5^{2} \cdot 2)^{\frac{1}{2}}

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

:فعّل قانون القوى

= 2^{\frac{1}{2}} (5^{2})^{\frac{1}{2}}

:فعّل قانون القوى

(5^{2})^{\frac{1}{2}} = 5^{\frac{2}{2}} = 5

= 5 \cdot 2^{\frac{1}{2}}

= \frac{98 1 ^{\frac{1}{2}}}{5 \cdot 2 ^{\frac{1}{2}}}

98 1^{\frac{1}{2}} = 3 \cdot 10 9^{\frac{1}{2}}

98 1^{\frac{1}{2}}

981

تحليل لعوامل أوّليّة لـ:

3^{2} \cdot 109

981

981 = 327 \cdot 3, 3

ينقسم على

981

= 3 \cdot 327

327 = 109 \cdot 3, 3

ينقسم على

327

= 3 \cdot 3 \cdot 109

مركّب من أعداد أوّليّة فقط، لذلك تحليل إضافيّ غير ممكن

3, 109

= 3 \cdot 3 \cdot 109

= 3^{2} \cdot 109

= (3^{2} \cdot 109)^{\frac{1}{2}}

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

:فعّل قانون القوى

= 10 9^{\frac{1}{2}} (3^{2})^{\frac{1}{2}}

:فعّل قانون القوى

(3^{2})^{\frac{1}{2}} = 3^{\frac{2}{2}} = 3

= 3 \cdot 10 9^{\frac{1}{2}}

= \frac{3 \cdot 10 9 ^{\frac{1}{2}}}{5 \cdot 2 ^{\frac{1}{2}}}

= 1 \cdot \frac{8}{15} \cdot \frac{29}{50} \cdot \frac{3 \cdot 10 9 ^{\frac{1}{2}}}{5 \cdot 2 ^{\frac{1}{2}}} (\frac{1}{10})^{\frac{5}{2}}

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

:اضرب كسور

= 1 \cdot \frac{8 \cdot 29 \cdot 3 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{15 \cdot 50 \cdot 5 \cdot 2 ^{\frac{1}{2}}}

\frac{8 \cdot 29 \cdot 3 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{15 \cdot 50 \cdot 5 \cdot 2 ^{\frac{1}{2}}} = \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

\frac{8 \cdot 29 \cdot 3 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{15 \cdot 50 \cdot 5 \cdot 2 ^{\frac{1}{2}}}

8 \cdot 29 \cdot 3 = 696 :

اضرب الأعداد

= \frac{696 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{15 \cdot 50 \cdot 5 \cdot 2 ^{\frac{1}{2}}}

15 \cdot 50 \cdot 5 = 3750 :

اضرب الأعداد

= \frac{696 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{3750 \cdot 2 ^{\frac{1}{2}}}

6 :

إلغ العوامل المشتركة

= \frac{116 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625 \cdot 2 ^{\frac{1}{2}}}

116

حلل إلى عوامل:

2^{2} \cdot 29

116 = 2^{2} \cdot 29

حلّل إلى عوامل

= \frac{2 ^{2} \cdot 29 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625 \cdot 2 ^{\frac{1}{2}}}

\frac{2 ^{2} \cdot 29 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625 \cdot 2 ^{\frac{1}{2}}}

اختزل:

\frac{29 \cdot 2 ^{\frac{3}{2}} \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625}

\frac{2 ^{2} \cdot 29 \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625 \cdot 2 ^{\frac{1}{2}}}

\frac{x ^{a}}{x ^{b}} = x^{a - b}

:فعّل قانون القوى

\frac{2 ^{2}}{2 ^{\frac{1}{2}}} = 2^{2 - \frac{1}{2}}

= \frac{29 \cdot 10 9 ^{\frac{1}{2}} \cdot 2 ^{- \frac{1}{2} + 2} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625}

2 - \frac{1}{2} = \frac{3}{2} :

اطرح الأعداد

= \frac{29 \cdot 2 ^{\frac{3}{2}} \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625}

= \frac{29 \cdot 2 ^{\frac{3}{2}} \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625}

2^{\frac{3}{2}} = 2 \cdot 2^{\frac{1}{2}}

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

x^{a + b} = x^{a} x^{b}

:فعّل قانون القوى

= 2^{1} \cdot 2^{\frac{1}{2}}

بسّط

= 2 \cdot 2^{\frac{1}{2}}

= \frac{29 \cdot 2 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}} ( \frac{1}{10} ) ^{\frac{5}{2}}}{625}

(\frac{1}{10})^{\frac{5}{2}} = \frac{1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

(\frac{1}{10})^{\frac{5}{2}}

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

:فعّل قانون القوى

= \frac{1 ^{\frac{5}{2}}}{1 0 ^{\frac{5}{2}}}

1 0^{\frac{5}{2}} = 1 0^{2} \cdot 1 0^{\frac{1}{2}}

1 0^{\frac{5}{2}}

1 0^{\frac{5}{2}} = 1 0^{2 + \frac{1}{2}}

= 1 0^{2 + \frac{1}{2}}

x^{a + b} = x^{a} x^{b}

:فعّل قانون القوى

= 1 0^{2} \cdot 1 0^{\frac{1}{2}}

= 1 0^{2} \cdot 1 0^{\frac{1}{2}}

1^{\frac{5}{2}} = 1^{2} \cdot 1^{\frac{1}{2}}

1^{\frac{5}{2}}

1^{\frac{5}{2}} = 1^{2 + \frac{1}{2}}

= 1^{2 + \frac{1}{2}}

x^{a + b} = x^{a} x^{b}

:فعّل قانون القوى

= 1^{2} \cdot 1^{\frac{1}{2}}

= \frac{1 ^{2} \cdot 1 ^{\frac{1}{2}}}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

1^{a} = 1

فعّل القانون

1^{2} = 1, 1^{\frac{1}{2}} = 1

= \frac{1 \cdot 1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

1 \cdot 1 = 1 :

اضرب الأعداد

= \frac{1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

= \frac{29 \cdot 2 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}} \cdot \frac{1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}}{625}

29 \cdot 2 = 58 :

اضرب الأعداد

= \frac{58 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}} \cdot \frac{1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}}{625}

58 \cdot 2^{\frac{1}{2}} \cdot 10 9^{\frac{1}{2}} \cdot \frac{1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

اضرب بـ:

\frac{29 \cdot 10 9 ^{\frac{1}{2}}}{50 \cdot 5 ^{\frac{1}{2}}}

58 \cdot 2^{\frac{1}{2}} \cdot 10 9^{\frac{1}{2}} \cdot \frac{1}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

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

:اضرب كسور

= \frac{1 \cdot 58 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}}}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

1 \cdot 58 = 58 :

اضرب الأعداد

= \frac{58 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}}}{1 0 ^{2} \cdot 1 0 ^{\frac{1}{2}}}

1 0^{2}

حلل إلى عوامل:

2^{2} \cdot 5^{2}

10 = 2 \cdot 5

حلّل إلى عوامل

= (2 \cdot 5)^{2}

(ab)^{c} = a^{c} b^{c}

:فعّل قانون القوى

= 2^{2} \cdot 5^{2}

1 0^{\frac{1}{2}}

حلل إلى عوامل:

2^{\frac{1}{2}} \cdot 5^{\frac{1}{2}}

10 = 2 \cdot 5

حلّل إلى عوامل

= (2 \cdot 5)^{\frac{1}{2}}

(ab)^{c} = a^{c} b^{c}

:فعّل قانون القوى

= 2^{\frac{1}{2}} \cdot 5^{\frac{1}{2}}

= \frac{58 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}}}{2 ^{2} \cdot 5 ^{2} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}}}

\frac{58 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}}}{2 ^{2} \cdot 5 ^{2} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}}}

اختزل:

\frac{58 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 2 ^{\frac{3}{2}} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}}}

\frac{58 \cdot 2 ^{\frac{1}{2}} \cdot 10 9 ^{\frac{1}{2}}}{2 ^{2} \cdot 5 ^{2} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}}}

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

:فعّل قانون القوى

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{58 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}} \cdot 2 ^{- \frac{1}{2} + 2}}

2 - \frac{1}{2} = \frac{3}{2} :

اطرح الأعداد

= \frac{58 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 2 ^{\frac{3}{2}} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}}}

= \frac{58 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 2 ^{\frac{3}{2}} \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}}}

5^{2} \cdot 2^{\frac{3}{2}} \cdot 2^{\frac{1}{2}} \cdot 5^{\frac{1}{2}} = 5^{2} \cdot 5^{\frac{1}{2}} \cdot 2^{\frac{3}{2} + \frac{1}{2}}

5^{2} \cdot 2^{\frac{3}{2}} \cdot 2^{\frac{1}{2}} \cdot 5^{\frac{1}{2}}

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

:فعّل قانون القوى

2^{\frac{3}{2}} \cdot 2^{\frac{1}{2}} = 2^{\frac{3}{2} + \frac{1}{2}}

= 5^{2} \cdot 2^{\frac{3}{2} + \frac{1}{2}} \cdot 5^{\frac{1}{2}}

= \frac{58 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 5 ^{\frac{1}{2}} \cdot 2 ^{\frac{3}{2} + \frac{1}{2}}}

58

حلل إلى عوامل:

2 \cdot 29

58 = 2 \cdot 29

حلّل إلى عوامل

= \frac{2 \cdot 29 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 5 ^{\frac{1}{2}} \cdot 2 ^{\frac{3}{2} + \frac{1}{2}}}

2 :

إلغ العوامل المشتركة

= \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{5 ^{2} \cdot 2 \cdot 5 ^{\frac{1}{2}}}

5^{2} \cdot 2 \cdot 5^{\frac{1}{2}} = 50 \cdot 5^{\frac{1}{2}}

5^{2} \cdot 2 \cdot 5^{\frac{1}{2}}

5^{2} = 25

= 25 \cdot 2 \cdot 5^{\frac{1}{2}}

25 \cdot 2 = 50 :

اضرب الأعداد

= 50 \cdot 5^{\frac{1}{2}}

= \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{50 \cdot 5 ^{\frac{1}{2}}}

= \frac{\frac{29 \cdot 10 9 ^{\frac{1}{2}}}{50 \cdot 5 ^{\frac{1}{2}}}}{625}

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

: استخدم ميزات الكسور التالية

= \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{50 \cdot 5 ^{\frac{1}{2}} \cdot 625}

50 \cdot 625 = 31250 :

اضرب الأعداد

= \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

= 1 \cdot \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

1 \cdot \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}} = \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}} :

اضرب

= \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

= \frac{29 \cdot 10 9 ^{\frac{1}{2}}}{31250 \cdot 5 ^{\frac{1}{2}}}

أمثلة شائعة

8sin(24)

8 sin (2 4^{\circ})

1/(sin^2(pi/4))

\frac{1}{sin ^{2} ( \frac{π}{4} )}

cos^2(24)-sin^2(24)

cos^{2} (2 4^{\circ}) - sin^{2} (2 4^{\circ})

arcsin(sqrt(1))

arcsin (1)

arccos(-3/(sqrt(14)))

arccos (- \frac{3}{14})

8/15*0.58*tan(45)*sqrt(2*9.81)*0.1^{5/2}

الحلّ

الحلّ

أمثلة شائعة

8/150.58tan(45)sqrt(29.81)*0.1^{5/2}