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شائع علم المثلثات >

sin^3(270)+csc^2(495)

الحلّ

sin^{3} (27 0^{\circ}) + csc^{2} (49 5^{\circ})

الحلّ

1

خطوات الحلّ

sin^{3} (27 0^{\circ}) + csc^{2} (49 5^{\circ})

Rewrite using trig identities:

sin (27 0^{\circ}) = 2 sin (13 5^{\circ}) cos (13 5^{\circ})

sin (27 0^{\circ})

sin (2 \cdot 13 5^{\circ})

كـ

sin (27 0^{\circ})

أكتب

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

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

:فعّل متطابقة الزاوية المضاعفة

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

= (2 sin (13 5^{\circ}) cos (13 5^{\circ}))^{3} + csc^{2} (49 5^{\circ})

بسّط

= 8 sin^{3} (13 5^{\circ}) cos^{3} (13 5^{\circ}) + csc^{2} (49 5^{\circ})

csc (49 5^{\circ}) = csc (13 5^{\circ})

csc (49 5^{\circ})

36 0^{\circ} + 13 5^{\circ}

كـ

49 5^{\circ}

اكتب مجددًا

= csc (36 0^{\circ} + 13 5^{\circ})

csc (x + 36 0^{\circ}) = csc (x)

csc :

استخدم دوريّة الـ

csc (36 0^{\circ} + 13 5^{\circ}) = csc (13 5^{\circ})

= csc (13 5^{\circ})

= 8 sin^{3} (13 5^{\circ}) cos^{3} (13 5^{\circ}) + csc^{2} (13 5^{\circ})

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:

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}

Rewrite using trig identities:

csc (13 5^{\circ}) = 2

csc (13 5^{\circ})

Rewrite using trig identities:

\frac{1}{sin ( 13 5 ^{\circ} )}

csc (13 5^{\circ})

csc (x) = \frac{1}{sin ( x )}

:Use the basic trigonometric identity

= \frac{1}{sin ( 13 5 ^{\circ} )}

= \frac{1}{sin ( 13 5 ^{\circ} )}

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}

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

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

بسّط:

2

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

\frac{1}{\frac{b}{c}} = \frac{c}{b}

: استخدم ميزات الكسور التالية

= \frac{2}{2}

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

:فعْل قانون الجذور

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

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

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

:فعّل قانون القوى

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

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

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

اطرح الأعداد

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

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

:فعْل قانون الجذور

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

= 2

= 2

= 8 (\frac{2}{2})^{3} (- \frac{2}{2})^{3} + (2)^{2}

8 (\frac{2}{2})^{3} (- \frac{2}{2})^{3} + (2)^{2}

بسّط:

1

8 (\frac{2}{2})^{3} (- \frac{2}{2})^{3} + (2)^{2}

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

8 (\frac{2}{2})^{3} (- \frac{2}{2})^{3}

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

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

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

:فعّل قانون القوى

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

(2)^{3} : 2^{\frac{3}{2}}

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

:فعْل قانون الجذور

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

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

:فعّل قانون القوى

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

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

\frac{1}{2} \cdot 3

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

:اضرب كسور

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

1 \cdot 3 = 3 :

اضرب الأعداد

= \frac{3}{2}

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

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

2^{\frac{3}{2}} = 22

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

= 22

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

2 :

إلغ العوامل المشتركة

= \frac{2}{2 ^{2}}

= 8 (- \frac{2}{2})^{3} \frac{2}{2 ^{2}}

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

(- \frac{2}{2})^{3}

فرديّ

n

إذا تحقّق أنّ

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

:فعّل قانون القوى

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

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

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

:فعّل قانون القوى

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

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

(2)^{3} : 2^{\frac{3}{2}}

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

:فعْل قانون الجذور

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

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

:فعّل قانون القوى

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

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

\frac{1}{2} \cdot 3

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

:اضرب كسور

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

1 \cdot 3 = 3 :

اضرب الأعداد

= \frac{3}{2}

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

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

2^{\frac{3}{2}} = 22

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

= 22

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

2 :

إلغ العوامل المشتركة

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

= 8 \cdot \frac{2}{2 ^{2}} (- \frac{2}{2 ^{2}})

(- a) = - a

:احذف الأقواس

= - 8 \cdot \frac{2}{2 ^{2}} \cdot \frac{2}{2 ^{2}}

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

:اضرب كسور

= - \frac{2 2 \cdot 8}{2 ^{2} \cdot 2 ^{2}}

22 \cdot 8 = 16

22 \cdot 8

a a = a

:فعْل قانون الجذور

22 = 2

= 2 \cdot 8

2 \cdot 8 = 16 :

اضرب الأعداد

= 16

= - \frac{16}{2 ^{2} \cdot 2 ^{2}}

2^{2} \cdot 2^{2} = 2^{4}

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

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

:فعّل قانون القوى

2^{2} \cdot 2^{2} = 2^{2 + 2}

= 2^{2 + 2}

2 + 2 = 4 :

اجمع الأعداد

= 2^{4}

= - \frac{16}{2 ^{4}}

\frac{16}{2 ^{4}}

اختزل:

1

\frac{16}{2 ^{4}}

16

حلل إلى عوامل:

2^{4}

16 = 2^{4}

حلّل إلى عوامل

= \frac{2 ^{4}}{2 ^{4}}

2^{4} :

إلغ العوامل المشتركة

= 1

= - 1

= - 1 + (2)^{2}

(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

= - 1 + 2

- 1 + 2 = 1 :

اطرح/اجمع الأعداد

= 1

= 1

أمثلة شائعة

cos(53.1)

cos (53. 1^{\circ})

(12.8sin(58))/(11.4)

\frac{12.8 sin ( 5 8 ^{\circ} )}{11.4}

cos^2(53)

cos^{2} (5 3^{\circ})

cos(-1/3)

cos (- \frac{1}{3})

sec((19pi)/6)

sec (\frac{19 π}{6})