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شائع علم المثلثات >

sin(10)=2sin(a)cos(a)

الحلّ

sin (1 0^{\circ}) = 2 sin (a) cos (a)

الحلّ

a = 5^{\circ} + 18 0^{\circ} n, a = 9 0^{\circ} - 5^{\circ} + 18 0^{\circ} n

راديان

a = \frac{π}{36} + πn, a = \frac{π}{2} - \frac{π}{36} + πn

خطوات الحلّ

sin (1 0^{\circ}) = 2 sin (a) cos (a)

من الطرفين

2 sin (a) cos (a)

اطرح

sin (1 0^{\circ}) - 2 sin (a) cos (a) = 0

Rewrite using trig identities

sin (1 0^{\circ}) - 2 sin (a) cos (a)

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

:فعّل متطابقة الزاوية المضاعفة

= sin (1 0^{\circ}) - sin (2 a)

sin (1 0^{\circ}) - sin (2 a) = 0

انقل

sin (1 0^{\circ})

إلى الجانب الأيمن

sin (1 0^{\circ}) - sin (2 a) = 0

من الطرفين

sin (1 0^{\circ})

اطرح

sin (1 0^{\circ}) - sin (2 a) - sin (1 0^{\circ}) = 0 - sin (1 0^{\circ})

بسّط

- sin (2 a) = - sin (1 0^{\circ})

- sin (2 a) = - sin (1 0^{\circ})

- 1

اقسم الطرفين على

- sin (2 a) = - sin (1 0^{\circ})

- 1

اقسم الطرفين على

\frac{- sin ( 2 a )}{- 1} = \frac{- sin ( 1 0 ^{\circ} )}{- 1}

بسّط

\frac{- sin ( 2 a )}{- 1} = \frac{- sin ( 1 0 ^{\circ} )}{- 1}

\frac{- sin ( 2 a )}{- 1}

بسّط:

sin (2 a)

\frac{- sin ( 2 a )}{- 1}

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

: استخدم ميزات الكسور التالية

= \frac{sin ( 2 a )}{1}

\frac{a}{1} = a

فعّل القانون

= sin (2 a)

\frac{- sin ( 1 0 ^{\circ} )}{- 1}

بسّط:

sin (1 0^{\circ})

\frac{- sin ( 1 0 ^{\circ} )}{- 1}

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

: استخدم ميزات الكسور التالية

= \frac{sin ( 1 0 ^{\circ} )}{1}

\frac{a}{1} = a

: استخدم ميزات الكسور التالية

= sin (1 0^{\circ})

sin (2 a) = sin (1 0^{\circ})

sin (2 a) = sin (1 0^{\circ})

sin (2 a) = sin (1 0^{\circ})

Apply trig inverse properties

sin (2 a) = sin (1 0^{\circ})

sin (2 a) = sin (1 0^{\circ}) :

حلول عامّة لـ

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

2 a = arcsin (sin (1 0^{\circ})) + 36 0^{\circ} n, 2 a = 18 0^{\circ} - arcsin (sin (1 0^{\circ})) + 36 0^{\circ} n

2 a = arcsin (sin (1 0^{\circ})) + 36 0^{\circ} n, 2 a = 18 0^{\circ} - arcsin (sin (1 0^{\circ})) + 36 0^{\circ} n

2 a = arcsin (sin (1 0^{\circ})) + 36 0^{\circ} n

حلّ:

a = 5^{\circ} + 18 0^{\circ} n

2 a = arcsin (sin (1 0^{\circ})) + 36 0^{\circ} n

2

اقسم الطرفين على

2 a = arcsin (sin (1 0^{\circ})) + 36 0^{\circ} n

2

اقسم الطرفين على

\frac{2 a}{2} = \frac{arcsin ( sin ( 1 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

بسّط

\frac{2 a}{2} = \frac{arcsin ( sin ( 1 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

\frac{2 a}{2}

بسّط:

a

\frac{2 a}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= a

\frac{arcsin ( sin ( 1 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

بسّط:

5^{\circ} + 18 0^{\circ} n

\frac{arcsin ( sin ( 1 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

\frac{arcsin ( sin ( 1 0 ^{\circ} ) )}{2} = 5^{\circ}

\frac{arcsin ( sin ( 1 0 ^{\circ} ) )}{2}

arcsin (sin (1 0^{\circ})) = 1 0^{\circ}

arcsin (sin (1 0^{\circ}))

a rcs in (s in (x)) = x, - 9 0^{\circ} \leq a \leq 9 0^{\circ}

من أجل

- 9 0^{\circ} \leq 1 0^{\circ} \leq 9 0^{\circ}

= 1 0^{\circ}

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

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

: استخدم ميزات الكسور التالية

= \frac{18 0 ^{\circ}}{18 \cdot 2}

18 \cdot 2 = 36 :

اضرب الأعداد

= 5^{\circ}

= 5^{\circ} + \frac{36 0 ^{\circ} n}{2}

\frac{36 0 ^{\circ} n}{2} = 18 0^{\circ} n

\frac{36 0 ^{\circ} n}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= 18 0^{\circ} n

= 5^{\circ} + 18 0^{\circ} n

a = 5^{\circ} + 18 0^{\circ} n

a = 5^{\circ} + 18 0^{\circ} n

a = 5^{\circ} + 18 0^{\circ} n

2 a = 18 0^{\circ} - arcsin (sin (1 0^{\circ})) + 36 0^{\circ} n

حلّ:

a = 9 0^{\circ} - 5^{\circ} + 18 0^{\circ} n

2 a = 18 0^{\circ} - arcsin (sin (1 0^{\circ})) + 36 0^{\circ} n

2

اقسم الطرفين على

2 a = 18 0^{\circ} - arcsin (sin (1 0^{\circ})) + 36 0^{\circ} n

2

اقسم الطرفين على

\frac{2 a}{2} = 9 0^{\circ} - \frac{arcsin ( sin ( 1 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

بسّط

\frac{2 a}{2} = 9 0^{\circ} - \frac{arcsin ( sin ( 1 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

\frac{2 a}{2}

بسّط:

a

\frac{2 a}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= a

9 0^{\circ} - \frac{arcsin ( sin ( 1 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

بسّط:

9 0^{\circ} - 5^{\circ} + 18 0^{\circ} n

9 0^{\circ} - \frac{arcsin ( sin ( 1 0 ^{\circ} ) )}{2} + \frac{36 0 ^{\circ} n}{2}

\frac{arcsin ( sin ( 1 0 ^{\circ} ) )}{2} = 5^{\circ}

\frac{arcsin ( sin ( 1 0 ^{\circ} ) )}{2}

arcsin (sin (1 0^{\circ})) = 1 0^{\circ}

arcsin (sin (1 0^{\circ}))

a rcs in (s in (x)) = x, - 9 0^{\circ} \leq a \leq 9 0^{\circ}

من أجل

- 9 0^{\circ} \leq 1 0^{\circ} \leq 9 0^{\circ}

= 1 0^{\circ}

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

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

: استخدم ميزات الكسور التالية

= \frac{18 0 ^{\circ}}{18 \cdot 2}

18 \cdot 2 = 36 :

اضرب الأعداد

= 5^{\circ}

= 9 0^{\circ} - 5^{\circ} + \frac{36 0 ^{\circ} n}{2}

\frac{36 0 ^{\circ} n}{2} = 18 0^{\circ} n

\frac{36 0 ^{\circ} n}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= 18 0^{\circ} n

= 9 0^{\circ} - 5^{\circ} + 18 0^{\circ} n

a = 9 0^{\circ} - 5^{\circ} + 18 0^{\circ} n

a = 9 0^{\circ} - 5^{\circ} + 18 0^{\circ} n

a = 9 0^{\circ} - 5^{\circ} + 18 0^{\circ} n

a = 5^{\circ} + 18 0^{\circ} n, a = 9 0^{\circ} - 5^{\circ} + 18 0^{\circ} n

رسم

أمثلة شائعة

cos^2(x)=2+2sin(x),0<= x<= 2pi

cos^{2} (x) = 2 + 2 sin (x), 0 \leq x \leq 2 π

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cos^{2} (x) = \frac{3}{5}

1-sec^2(x)=tan^2(x)

1 - sec^{2} (x) = tan^{2} (x)

csc^2(x)=(1/(cos(x)))^2

csc^{2} (x) = (\frac{1}{cos ( x )})^{2}

-2cos^2(x)+3cos(x)=1

- 2 cos^{2} (x) + 3 cos (x) = 1