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Popular >

solvefor t,x-4=sec^2(t)

Solution

solve for

t, x - 4 = sec^{2} (t)

Solution

t = arcsec (x - 4) + 2 πn, t = - arcsec (x - 4) + 2 πn, t = arcsec (- x - 4) + 2 πn, t = - arcsec (- x - 4) + 2 πn

Solution steps

x - 4 = sec^{2} (t)

Switch sides

sec^{2} (t) = x - 4

Solve by substitution

sec^{2} (t) = x - 4

Let:

sec (t) = u

u^{2} = x - 4

u^{2} = x - 4 : u = x - 4, u = - x - 4

u^{2} = x - 4

For

x^{2} = f (a)

the solutions are

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

u = x - 4, u = - x - 4

Substitute back

u = sec (t)

sec (t) = x - 4, sec (t) = - x - 4

sec (t) = x - 4, sec (t) = - x - 4

sec (t) = x - 4 : t = arcsec (x - 4) + 2 πn, t = - arcsec (x - 4) + 2 πn

sec (t) = x - 4

Apply trig inverse properties

sec (t) = x - 4

General solutions for

sec (t) = x - 4

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

t = arcsec (x - 4) + 2 πn, t = - arcsec (x - 4) + 2 πn

t = arcsec (x - 4) + 2 πn, t = - arcsec (x - 4) + 2 πn

sec (t) = - x - 4 : t = arcsec (- x - 4) + 2 πn, t = - arcsec (- x - 4) + 2 πn

sec (t) = - x - 4

Apply trig inverse properties

sec (t) = - x - 4

General solutions for

sec (t) = - x - 4

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

t = arcsec (- x - 4) + 2 πn, t = - arcsec (- x - 4) + 2 πn

t = arcsec (- x - 4) + 2 πn, t = - arcsec (- x - 4) + 2 πn

Combine all the solutions

t = arcsec (x - 4) + 2 πn, t = - arcsec (x - 4) + 2 πn, t = arcsec (- x - 4) + 2 πn, t = - arcsec (- x - 4) + 2 πn

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