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شائع علم المثلثات >

solvefor x,y=(sec(x))/(cos^1(x))+8y^2

الحلّ

solve for

x, y = \frac{sec ( x )}{cos ^{1} ( x )} + 8 y^{2}

الحلّ

x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn, x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

خطوات الحلّ

y = \frac{sec ( x )}{cos ^{1} ( x )} + 8 y^{2}

بدّل الأطراف

\frac{sec ( x )}{cos ^{1} ( x )} + 8 y^{2} = y

من الطرفين

y

اطرح

\frac{sec ( x )}{cos ( x )} + 8 y^{2} - y = 0

\frac{sec ( x )}{cos ( x )} + 8 y^{2} - y

بسّط:

\frac{sec ( x ) + 8 y ^{2} cos ( x ) - y cos ( x )}{cos ( x )}

\frac{sec ( x )}{cos ( x )} + 8 y^{2} - y

8 y^{2} = \frac{8 y ^{2} cos ( x )}{cos ( x )}, y = \frac{y cos ( x )}{cos ( x )}

:حوّل الأعداد لكسور

= \frac{sec ( x )}{cos ( x )} + \frac{8 y ^{2} cos ( x )}{cos ( x )} - \frac{y cos ( x )}{cos ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{sec ( x ) + 8 y ^{2} cos ( x ) - y cos ( x )}{cos ( x )}

\frac{sec ( x ) + 8 y ^{2} cos ( x ) - y cos ( x )}{cos ( x )} = 0

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

sec (x) + 8 y^{2} cos (x) - y cos (x) = 0

Rewrite using trig identities

sec (x) - cos (x) y + 8 cos (x) y^{2}

cos (x) = \frac{1}{sec ( x )}

:Use the basic trigonometric identity

= sec (x) - \frac{1}{sec ( x )} y + 8 \cdot \frac{1}{sec ( x )} y^{2}

sec (x) - \frac{1}{sec ( x )} y + 8 \cdot \frac{1}{sec ( x )} y^{2}

بسّط:

sec (x) + \frac{- y + 8 y ^{2}}{sec ( x )}

sec (x) - \frac{1}{sec ( x )} y + 8 \cdot \frac{1}{sec ( x )} y^{2}

\frac{1}{sec ( x )} y = \frac{y}{sec ( x )}

\frac{1}{sec ( x )} y

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

:اضرب كسور

= \frac{1 \cdot y}{sec ( x )}

1 \cdot y = y :

اضرب

= \frac{y}{sec ( x )}

8 \cdot \frac{1}{sec ( x )} y^{2} = \frac{8 y ^{2}}{sec ( x )}

8 \cdot \frac{1}{sec ( x )} y^{2}

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

:اضرب كسور

= \frac{1 \cdot 8 y ^{2}}{sec ( x )}

1 \cdot 8 = 8 :

اضرب الأعداد

= \frac{8 y ^{2}}{sec ( x )}

= sec (x) - \frac{y}{sec ( x )} + \frac{8 y ^{2}}{sec ( x )}

- \frac{y}{sec ( x )} + \frac{8 y ^{2}}{sec ( x )}

وحّد الكسور:

\frac{- y + 8 y ^{2}}{sec ( x )}

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

فعّل القانون

= \frac{- y + 8 y ^{2}}{sec ( x )}

= sec (x) + \frac{8 y ^{2} - y}{sec ( x )}

= sec (x) + \frac{- y + 8 y ^{2}}{sec ( x )}

\frac{- y + 8 y ^{2}}{sec ( x )} + sec (x) = 0

بالاستعانة بطريقة التعويض

\frac{- y + 8 y ^{2}}{sec ( x )} + sec (x) = 0

sec (x) = u :

على افتراض أنّ

\frac{- y + 8 y ^{2}}{u} + u = 0

\frac{- y + 8 y ^{2}}{u} + u = 0 : u = y - 8 y^{2}, u = - y - 8 y^{2}

\frac{- y + 8 y ^{2}}{u} + u = 0

u

اضرب الطرفين بـ

\frac{- y + 8 y ^{2}}{u} + u = 0

u

اضرب الطرفين بـ

\frac{- y + 8 y ^{2}}{u} u + uu = 0 \cdot u

بسّط

\frac{- y + 8 y ^{2}}{u} u + uu = 0 \cdot u

\frac{- y + 8 y ^{2}}{u} u

بسّط:

- y + 8 y^{2}

\frac{- y + 8 y ^{2}}{u} u

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

:اضرب كسور

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

u :

إلغ العوامل المشتركة

= - - y + 8 y^{2}

uu

بسّط:

u^{2}

uu

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

:فعّل قانون القوى

uu = u^{1 + 1}

= u^{1 + 1}

1 + 1 = 2 :

اجمع الأعداد

= u^{2}

0 \cdot u

بسّط:

0

0 \cdot u

0 \cdot a = 0

فعّل القانون

= 0

- y + 8 y^{2} + u^{2} = 0

- y + 8 y^{2} + u^{2} = 0

- y + 8 y^{2} + u^{2} = 0

- y + 8 y^{2} + u^{2} = 0

حلّ:

u = y - 8 y^{2}, u = - y - 8 y^{2}

- y + 8 y^{2} + u^{2} = 0

انقل

y

إلى الجانب الأيمن

- y + 8 y^{2} + u^{2} = 0

للطرفين

y

أضف

- y + 8 y^{2} + u^{2} + y = 0 + y

بسّط

8 y^{2} + u^{2} = y

8 y^{2} + u^{2} = y

انقل

8 y^{2}

إلى الجانب الأيمن

8 y^{2} + u^{2} = y

من الطرفين

8 y^{2}

اطرح

8 y^{2} + u^{2} - 8 y^{2} = y - 8 y^{2}

بسّط

u^{2} = y - 8 y^{2}

u^{2} = y - 8 y^{2}

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

الحلول هي

x^{2} = f (a)

لـ

u = y - 8 y^{2}, u = - y - 8 y^{2}

u = y - 8 y^{2}, u = - y - 8 y^{2}

u = sec (x)

استبدل مجددًا

sec (x) = y - 8 y^{2}, sec (x) = - y - 8 y^{2}

sec (x) = y - 8 y^{2}, sec (x) = - y - 8 y^{2}

sec (x) = y - 8 y^{2} : x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn

sec (x) = y - 8 y^{2}

Apply trig inverse properties

sec (x) = y - 8 y^{2}

sec (x) = y - 8 y^{2} :

حلول عامّة لـ

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn

x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn

sec (x) = - y - 8 y^{2} : x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

sec (x) = - y - 8 y^{2}

Apply trig inverse properties

sec (x) = - y - 8 y^{2}

sec (x) = - y - 8 y^{2} :

حلول عامّة لـ

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

وحّد الحلول

x = arcsec (y - 8 y^{2}) + 2 πn, x = - arcsec (y - 8 y^{2}) + 2 πn, x = arcsec (- y - 8 y^{2}) + 2 πn, x = - arcsec (- y - 8 y^{2}) + 2 πn

رسم

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