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Popular Trigonometry >

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

Solution

solve for

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

Solution

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

Solution steps

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

Switch sides

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

Subtract

y

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

= \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}

Use the basic trigonometric identity:

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

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

Simplify

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

Multiply fractions:

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

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

Multiply:

1 \cdot y = y

= \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}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 8 = 8

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

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

Combine the fractions

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

Apply rule

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

= \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

Solve by substitution

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

Let:

sec (x) = u

\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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

u

= - - y + 8 y^{2}

Simplify

uu : u^{2}

uu

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

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

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

Move

y

to the right side

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

Add

y

to both sides

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

Simplify

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

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

Move

8 y^{2}

to the right side

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

Subtract

8 y^{2}

from both sides

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

Simplify

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

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

For

x^{2} = f (a)

the solutions are

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

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

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

Substitute back

u = sec (x)

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}

General solutions for

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

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}

General solutions for

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

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

Combine all the solutions

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

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