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شائع علم المثلثات >

2cos^2(165)-1

الحلّ

2 cos^{2} (16 5^{\circ}) - 1

الحلّ

\frac{2 + 3}{2} - 1

عشري

0.86602 \dots

خطوات الحلّ

2 cos^{2} (16 5^{\circ}) - 1

Rewrite using trig identities:

cos (16 5^{\circ}) = \frac{- 6 - 2}{4}

cos (16 5^{\circ})

Rewrite using trig identities:

cos (13 5^{\circ}) cos (3 0^{\circ}) - sin (13 5^{\circ}) sin (3 0^{\circ})

cos (16 5^{\circ})

cos (13 5^{\circ} + 3 0^{\circ})

كـ

cos (16 5^{\circ})

أكتب

= cos (13 5^{\circ} + 3 0^{\circ})

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

:فعّل متطابقة الجمع لزوايا

= cos (13 5^{\circ}) cos (3 0^{\circ}) - sin (13 5^{\circ}) sin (3 0^{\circ})

= cos (13 5^{\circ}) cos (3 0^{\circ}) - sin (13 5^{\circ}) sin (3 0^{\circ})

استخدم المتطابقة الأساسيّة التالية:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 5^{\circ})

cos (x)

periodicity table with

36 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{2}{2}

استخدم المتطابقة الأساسيّة التالية:

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

cos (x)

periodicity table with

36 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

استخدم المتطابقة الأساسيّة التالية:

sin (13 5^{\circ}) = \frac{2}{2}

sin (13 5^{\circ})

sin (x)

periodicity table with

36 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

استخدم المتطابقة الأساسيّة التالية:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 0^{\circ})

sin (x)

periodicity table with

36 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= (- \frac{2}{2}) \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2}

(- \frac{2}{2}) \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2}

بسّط:

\frac{- 6 - 2}{4}

(- \frac{2}{2}) \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2}

(- a) = - a

:احذف الأقواس

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

\frac{2}{2} \cdot \frac{3}{2} = \frac{6}{4}

\frac{2}{2} \cdot \frac{3}{2}

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

:اضرب كسور

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

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{2 3}{4}

23

بسّط:

6

23

a b = a \cdot b

:فعْل قانون الجذور

23 = 2 \cdot 3

= 2 \cdot 3

2 \cdot 3 = 6 :

اضرب الأعداد

= 6

= \frac{6}{4}

\frac{2}{2} \cdot \frac{1}{2} = \frac{2}{4}

\frac{2}{2} \cdot \frac{1}{2}

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

:اضرب كسور

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

2 \cdot 1 = 2 :

اضرب

= \frac{2}{2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= \frac{2}{4}

= - \frac{6}{4} - \frac{2}{4}

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

فعّل القانون

= \frac{- 6 - 2}{4}

= \frac{- 6 - 2}{4}

= 2 (\frac{- 6 - 2}{4})^{2} - 1

2 (\frac{- 6 - 2}{4})^{2} - 1

بسّط:

\frac{2 + 3}{2} - 1

2 (\frac{- 6 - 2}{4})^{2} - 1

2 (\frac{- 6 - 2}{4})^{2} = \frac{2 + 3}{2}

2 (\frac{- 6 - 2}{4})^{2}

(\frac{- 6 - 2}{4})^{2} = \frac{2 + 3}{4}

(\frac{- 6 - 2}{4})^{2}

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

:فعّل قانون القوى

= \frac{( - 6 - 2 ) ^{2}}{4 ^{2}}

(- 6 - 2)^{2} = 8 + 43

(- 6 - 2)^{2}

(a - b)^{2} = a^{2} - 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

a = - 6, b = 2

= (- 6)^{2} - 2 (- 6) 2 + (2)^{2}

(- 6)^{2} - 2 (- 6) 2 + (2)^{2}

بسّط:

8 + 43

(- 6)^{2} - 2 (- 6) 2 + (2)^{2}

- (- a) = a

فعّل القانون

= (- 6)^{2} + 262 + (2)^{2}

(- 6)^{2} = 6

(- 6)^{2}

زوجيّ n

إذا تحقّق أنّ

, (- a)^{n} = a^{n}

:فعّل قانون القوى

(- 6)^{2} = (6)^{2}

= (6)^{2}

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

:فعْل قانون الجذور

= (6^{\frac{1}{2}})^{2}

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

:فعّل قانون القوى

= 6^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= 1

= 6

262 = 43

262

6 = 2 \cdot 3 :

حلّل العدد لعوامله الأوّليّة

= 2 2 \cdot 3 2

n ab = n a n b

:فعْل قانون الجذور

2 \cdot 3 = 23

= 2232

a a = a

:فعْل قانون الجذور

22 = 2

= 2 \cdot 23

2 \cdot 2 = 4 :

اضرب الأعداد

= 43

(2)^{2} = 2

(2)^{2}

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

:فعْل قانون الجذور

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

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

:فعّل قانون القوى

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

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

\frac{1}{2} \cdot 2

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= 1

= 2

= 6 + 43 + 2

6 + 2 = 8 :

اجمع الأعداد

= 8 + 43

= 8 + 43

= \frac{8 + 4 3}{4 ^{2}}

8 + 43

حلل إلى عوامل:

4 (2 + 3)

8 + 43

أعد الكتابة كـ

= 4 \cdot 2 + 43

4

قم باخراج العامل المشترك

= 4 (2 + 3)

= \frac{4 ( 2 + 3 )}{4 ^{2}}

4 :

إلغ العوامل المشتركة

= \frac{2 + 3}{4}

= 2 \cdot \frac{2 + 3}{4}

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

:اضرب كسور

= \frac{( 2 + 3 ) \cdot 2}{4}

2 :

إلغ العوامل المشتركة

= \frac{2 + 3}{2}

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

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

أمثلة شائعة

arccot(3/2)

arccot (\frac{3}{2})

sin(15)cos(45)+cos(15)sin(45)

sin (1 5^{\circ}) cos (4 5^{\circ}) + cos (1 5^{\circ}) sin (4 5^{\circ})

cos(50)cos(20)+sin(50)sin(20)

cos (5 0^{\circ}) cos (2 0^{\circ}) + sin (5 0^{\circ}) sin (2 0^{\circ})

(tan(45)+tan(30))/(1-tan(45)tan(30))

\frac{tan ( 4 5 ^{\circ} ) + tan ( 3 0 ^{\circ} )}{1 - tan ( 4 5 ^{\circ} ) tan ( 3 0 ^{\circ} )}

3sin((5pi)/4)

3 sin (\frac{5 π}{4})