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شائع علم المثلثات >

1/(csc^2(a))=1-sin^2(a)

الحلّ

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

الحلّ

a = \frac{π}{4} + 2 πn, a = \frac{3 π}{4} + 2 πn, a = \frac{5 π}{4} + 2 πn, a = \frac{7 π}{4} + 2 πn

درجات

a = 4 5^{\circ} + 36 0^{\circ} n, a = 13 5^{\circ} + 36 0^{\circ} n, a = 22 5^{\circ} + 36 0^{\circ} n, a = 31 5^{\circ} + 36 0^{\circ} n

خطوات الحلّ

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

من الطرفين

1 - sin^{2} (a)

اطرح

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

\frac{1}{csc ^{2} ( a )} - 1 + sin^{2} (a)

بسّط:

\frac{1 - csc ^{2} ( a ) + sin ^{2} ( a ) csc ^{2} ( a )}{csc ^{2} ( a )}

\frac{1}{csc ^{2} ( a )} - 1 + sin^{2} (a)

1 = \frac{1 csc ^{2} ( a )}{csc ^{2} ( a )}, sin^{2} (a) = \frac{sin ^{2} ( a ) csc ^{2} ( a )}{csc ^{2} ( a )}

:حوّل الأعداد لكسور

= \frac{1}{csc ^{2} ( a )} - \frac{1 \cdot csc ^{2} ( a )}{csc ^{2} ( a )} + \frac{sin ^{2} ( a ) csc ^{2} ( a )}{csc ^{2} ( a )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{1 - 1 \cdot csc ^{2} ( a ) + sin ^{2} ( a ) csc ^{2} ( a )}{csc ^{2} ( a )}

1 \cdot csc^{2} (a) = csc^{2} (a) :

اضرب

= \frac{1 - csc ^{2} ( a ) + sin ^{2} ( a ) csc ^{2} ( a )}{csc ^{2} ( a )}

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

1 - csc^{2} (a) + sin^{2} (a) csc^{2} (a) = 0

Rewrite using trig identities

1 - csc^{2} (a) + csc^{2} (a) sin^{2} (a)

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

:Use the basic trigonometric identity

= 1 - csc^{2} (a) + csc^{2} (a) (\frac{1}{csc ( a )})^{2}

1 - csc^{2} (a) + csc^{2} (a) (\frac{1}{csc ( a )})^{2}

بسّط:

- csc^{2} (a) + 2

1 - csc^{2} (a) + csc^{2} (a) (\frac{1}{csc ( a )})^{2}

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

csc^{2} (a) (\frac{1}{csc ( a )})^{2}

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

(\frac{1}{csc ( a )})^{2}

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

:فعّل قانون القوى

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

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

:اضرب كسور

= \frac{1 \cdot csc ^{2} ( a )}{csc ^{2} ( a )}

csc^{2} (a) :

إلغ العوامل المشتركة

= 1

= 1 - csc^{2} (a) + 1

جمّع التعابير المتشابهة

= - csc^{2} (a) + 1 + 1

1 + 1 = 2 :

اجمع الأعداد

= - csc^{2} (a) + 2

= - csc^{2} (a) + 2

2 - csc^{2} (a) = 0

بالاستعانة بطريقة التعويض

2 - csc^{2} (a) = 0

csc (a) = u :

على افتراض أنّ

2 - u^{2} = 0

2 - u^{2} = 0 : u = 2, u = - 2

2 - u^{2} = 0

انقل

2

إلى الجانب الأيمن

2 - u^{2} = 0

من الطرفين

2

اطرح

2 - u^{2} - 2 = 0 - 2

بسّط

- u^{2} = - 2

- u^{2} = - 2

- 1

اقسم الطرفين على

- u^{2} = - 2

- 1

اقسم الطرفين على

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

بسّط

u^{2} = 2

u^{2} = 2

x = f (a), - f (a)

الحلول هي

x^{2} = f (a)

لـ

u = 2, u = - 2

u = csc (a)

استبدل مجددًا

csc (a) = 2, csc (a) = - 2

csc (a) = 2, csc (a) = - 2

csc (a) = 2 : a = \frac{π}{4} + 2 πn, a = \frac{3 π}{4} + 2 πn

csc (a) = 2

csc (a) = 2 :

حلول عامّة لـ

csc (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

a = \frac{π}{4} + 2 πn, a = \frac{3 π}{4} + 2 πn

a = \frac{π}{4} + 2 πn, a = \frac{3 π}{4} + 2 πn

csc (a) = - 2 : a = \frac{5 π}{4} + 2 πn, a = \frac{7 π}{4} + 2 πn

csc (a) = - 2

csc (a) = - 2 :

حلول عامّة لـ

csc (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

a = \frac{5 π}{4} + 2 πn, a = \frac{7 π}{4} + 2 πn

a = \frac{5 π}{4} + 2 πn, a = \frac{7 π}{4} + 2 πn

وحّد الحلول

a = \frac{π}{4} + 2 πn, a = \frac{3 π}{4} + 2 πn, a = \frac{5 π}{4} + 2 πn, a = \frac{7 π}{4} + 2 πn

رسم

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