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

Solution

solve for

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

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 ) + 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

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

Solve by substitution

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} : 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

3 cot^{2} (θ) - 4 csc (θ) = 1

sin (2 x) sin (x) = 0

4 - sin (θ) = cos (2 θ)

sin (θ) = - 0.3

sin (x) = \frac{1}{1.52}