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Popular >

(tan(x))(sin^2(x)-a)(sec(x)-a)=0

Solution

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

Solution

x = πn, x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn, x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn, x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

Solution steps

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

Solving each part separately

tan (x) = 0 or sin^{2} (x) - a = 0 or sec (x) - a = 0

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

sin^{2} (x) - a = 0 : x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn, x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

sin^{2} (x) - a = 0

Solve by substitution

sin^{2} (x) - a = 0

Let:

sin (x) = u

u^{2} - a = 0

u^{2} - a = 0 : u = a, u = - a

u^{2} - a = 0

Move

a

to the right side

u^{2} - a = 0

Add

a

to both sides

u^{2} - a + a = 0 + a

Simplify

u^{2} = a

u^{2} = a

For

x^{2} = f (a)

the solutions are

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

u = a, u = - a

Substitute back

u = sin (x)

sin (x) = a, sin (x) = - a

sin (x) = a, sin (x) = - a

sin (x) = a : x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn

sin (x) = a

Apply trig inverse properties

sin (x) = a

General solutions for

sin (x) = a

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn

x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn

sin (x) = - a : x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

sin (x) = - a

Apply trig inverse properties

sin (x) = - a

General solutions for

sin (x) = - a

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

Combine all the solutions

x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn, x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

sec (x) - a = 0 : x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

sec (x) - a = 0

Move

a

to the right side

sec (x) - a = 0

Add

a

to both sides

sec (x) - a + a = 0 + a

Simplify

sec (x) = a

sec (x) = a

Apply trig inverse properties

sec (x) = a

General solutions for

sec (x) = a

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

x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

Combine all the solutions

x = πn, x = arcsin (a) + 2 πn, x = π + arcsin (- a) + 2 πn, x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn, x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

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