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شائع علم المثلثات >

sec^2(x)= 1/2 csc^2(x)+tan^2(x)

الحلّ

sec^{2} (x) = \frac{1}{2} csc^{2} (x) + tan^{2} (x)

الحلّ

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

درجات

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

خطوات الحلّ

sec^{2} (x) = \frac{1}{2} csc^{2} (x) + tan^{2} (x)

من الطرفين

\frac{1}{2} csc^{2} (x) + tan^{2} (x)

اطرح

sec^{2} (x) - \frac{1}{2} csc^{2} (x) - tan^{2} (x) = 0

sec^{2} (x) - \frac{1}{2} csc^{2} (x) - tan^{2} (x)

بسّط:

\frac{2 sec ^{2} ( x ) - csc ^{2} ( x ) - 2 tan ^{2} ( x )}{2}

sec^{2} (x) - \frac{1}{2} csc^{2} (x) - tan^{2} (x)

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

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

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

:اضرب كسور

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

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

اضرب

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

= sec^{2} (x) - \frac{csc ^{2} ( x )}{2} - tan^{2} (x)

sec^{2} (x) = \frac{sec ^{2} ( x ) 2}{2}, tan^{2} (x) = \frac{tan ^{2} ( x ) 2}{2}

:حوّل الأعداد لكسور

= \frac{sec ^{2} ( x ) \cdot 2}{2} - \frac{csc ^{2} ( x )}{2} - \frac{tan ^{2} ( x ) \cdot 2}{2}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{sec ^{2} ( x ) \cdot 2 - csc ^{2} ( x ) - tan ^{2} ( x ) \cdot 2}{2}

\frac{2 sec ^{2} ( x ) - csc ^{2} ( x ) - 2 tan ^{2} ( x )}{2} = 0

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

2 sec^{2} (x) - csc^{2} (x) - 2 tan^{2} (x) = 0

Rewrite using trig identities

- csc^{2} (x) + 2 sec^{2} (x) - 2 tan^{2} (x)

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

:فعّل نطريّة فيتاغوروس

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

= - csc^{2} (x) + 2 sec^{2} (x) - 2 (sec^{2} (x) - 1)

- csc^{2} (x) + 2 sec^{2} (x) - 2 (sec^{2} (x) - 1)

بسّط:

- csc^{2} (x) + 2

- csc^{2} (x) + 2 sec^{2} (x) - 2 (sec^{2} (x) - 1)

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

وسٌع:

- 2 sec^{2} (x) + 2

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

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = - 2, b = sec^{2} (x), c = 1

= - 2 sec^{2} (x) - (- 2) \cdot 1

فعّل قوانين سالب-موجب

- (- a) = a

= - 2 sec^{2} (x) + 2 \cdot 1

2 \cdot 1 = 2 :

اضرب الأعداد

= - 2 sec^{2} (x) + 2

= - csc^{2} (x) + 2 sec^{2} (x) - 2 sec^{2} (x) + 2

2 sec^{2} (x) - 2 sec^{2} (x) = 0 :

اجمع العناصر المتشابهة

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

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

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

بالاستعانة بطريقة التعويض

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

csc (x) = 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 (x)

استبدل مجددًا

csc (x) = 2, csc (x) = - 2

csc (x) = 2, csc (x) = - 2

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

csc (x) = 2

csc (x) = 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

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

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

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

csc (x) = - 2

csc (x) = - 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

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

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

وحّد الحلول

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

رسم

أمثلة شائعة

4cos(2θ)-3cos(θ)+4=-cos(θ)+3

4 cos (2 θ) - 3 cos (θ) + 4 = - cos (θ) + 3

cos(2x)=0.28

cos (2 x) = 0.28

cos(3x)=sqrt(3)sin(3x)

cos (3 x) = 3 sin (3 x)

-3cos(2x)=0

- 3 cos (2 x) = 0

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

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