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شائع علم المثلثات >

sin(2x)=(-1)/(sqrt(3))

الحلّ

sin (2 x) = \frac{- 1}{3}

الحلّ

x = - \frac{0.61547 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.61547 \dots}{2} + πn

+1

راديان

x = - 0.30773 \dots + πn, x = \frac{π}{2} + \frac{0.61547 \dots}{2} + πn

خطوات الحلّ

sin (2 x) = \frac{- 1}{3}

\frac{- 1}{3}

بسّط:

- \frac{3}{3}

\frac{- 1}{3}

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

: استخدم ميزات الكسور التالية

= - \frac{1}{3}

- \frac{1}{3}

حوّل لصيغة عدد كسريّ:

- \frac{3}{3}

- \frac{1}{3}

\frac{3}{3}

اضرب بالمرافق

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

a a = a

:فعْل قانون الجذور

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

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

Apply trig inverse properties

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

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

حلول عامّة لـ

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

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

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

2 x = arcsin (- \frac{3}{3}) + 2 πn

حلّ:

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

2 x = arcsin (- \frac{3}{3}) + 2 πn

arcsin (- \frac{3}{3}) + 2 πn

بسّط:

- arcsin (\frac{1}{3}) + 2 πn

arcsin (- \frac{3}{3}) + 2 πn

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

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

\frac{3}{3} = \frac{1}{3}

\frac{3}{3}

n a = a^{\frac{1}{n}}

:فعْل قانون الجذور

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

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

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

:فعّل قانون القوى

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

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

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

اطرح الأعداد

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

a^{\frac{1}{n}} = n a

:فعْل قانون الجذور

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

= \frac{1}{3}

= arcsin (- \frac{1}{3})

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

استخدم القانون التالي

arcsin (- \frac{1}{3}) = - arcsin (\frac{1}{3})

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

= - arcsin (\frac{3}{3}) + 2 πn

\frac{3}{3} = \frac{1}{3}

\frac{3}{3}

n a = a^{\frac{1}{n}}

:فعْل قانون الجذور

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

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

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

:فعّل قانون القوى

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

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

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

اطرح الأعداد

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

a^{\frac{1}{n}} = n a

:فعْل قانون الجذور

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

= \frac{1}{3}

= - arcsin (\frac{1}{3}) + 2 πn

2 x = - arcsin (\frac{1}{3}) + 2 πn

2

اقسم الطرفين على

2 x = - arcsin (\frac{1}{3}) + 2 πn

2

اقسم الطرفين على

\frac{2 x}{2} = - \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

بسّط

\frac{2 x}{2} = - \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

\frac{2 x}{2}

بسّط:

x

\frac{2 x}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= x

- \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

بسّط:

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

- \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= - \frac{arcsin ( \frac{1}{3} )}{2} + πn

arcsin (\frac{1}{3}) = arcsin (\frac{3}{3})

arcsin (\frac{1}{3})

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

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

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

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

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

2 x = π + arcsin (\frac{3}{3}) + 2 πn

حلّ:

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

2 x = π + arcsin (\frac{3}{3}) + 2 πn

π + arcsin (\frac{3}{3}) + 2 πn

بسّط:

π + arcsin (\frac{1}{3}) + 2 πn

π + arcsin (\frac{3}{3}) + 2 πn

\frac{3}{3} = \frac{1}{3}

\frac{3}{3}

n a = a^{\frac{1}{n}}

:فعْل قانون الجذور

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

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

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

:فعّل قانون القوى

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

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

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

اطرح الأعداد

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

a^{\frac{1}{n}} = n a

:فعْل قانون الجذور

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

= \frac{1}{3}

= π + arcsin (\frac{1}{3}) + 2 πn

2 x = π + arcsin (\frac{1}{3}) + 2 πn

2

اقسم الطرفين على

2 x = π + arcsin (\frac{1}{3}) + 2 πn

2

اقسم الطرفين على

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

بسّط

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

\frac{2 x}{2}

بسّط:

x

\frac{2 x}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= x

\frac{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

بسّط:

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

\frac{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + \frac{2 πn}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= \frac{π}{2} + \frac{arcsin ( \frac{1}{3} )}{2} + πn

arcsin (\frac{1}{3}) = arcsin (\frac{3}{3})

arcsin (\frac{1}{3})

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

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

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

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

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

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

أظهر الحلّ بالتمثيل العشريّ

x = - \frac{0.61547 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.61547 \dots}{2} + πn

رسم

أمثلة شائعة

cos(2θ)=1+3sin(θ)

cos (2 θ) = 1 + 3 sin (θ)

cos(2x)-cos(x)=-1,0<= x<2pi

cos (2 x) - cos (x) = - 1, 0 \leq x < 2 π

sin(x)=-1/(sqrt(10))

sin (x) = - \frac{1}{10}

sec^2(θ)+sec(θ)=2,2pi<= θ<= 3pi

sec^{2} (θ) + sec (θ) = 2, 2 π \leq θ \leq 3 π

6 sec^{2} (x) = 0