What is the general solution for -1-2sec^2(x)=-3sec^2(x) ?

The general solution for -1-2sec^2(x)=-3sec^2(x) is x=2pin,x=pi+2pin

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-1-2sec^2(x)=-3sec^2(x)

Solution

- 1 - 2 sec^{2} (x) = - 3 sec^{2} (x)

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

- 1 - 2 sec^{2} (x) = - 3 sec^{2} (x)

Solve by substitution

- 1 - 2 sec^{2} (x) = - 3 sec^{2} (x)

Let:

sec (x) = u

- 1 - 2 u^{2} = - 3 u^{2}

- 1 - 2 u^{2} = - 3 u^{2} : u = 1, u = - 1

- 1 - 2 u^{2} = - 3 u^{2}

Move

1

to the right side

- 1 - 2 u^{2} = - 3 u^{2}

Add

1

to both sides

- 1 - 2 u^{2} + 1 = - 3 u^{2} + 1

Simplify

- 2 u^{2} = - 3 u^{2} + 1

- 2 u^{2} = - 3 u^{2} + 1

Move

3 u^{2}

to the left side

- 2 u^{2} = - 3 u^{2} + 1

Add

3 u^{2}

to both sides

- 2 u^{2} + 3 u^{2} = - 3 u^{2} + 1 + 3 u^{2}

Simplify

u^{2} = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

Substitute back

u = sec (x)

sec (x) = 1, sec (x) = - 1

sec (x) = 1, sec (x) = - 1

sec (x) = 1 : x = 2 πn

sec (x) = 1

General solutions for

sec (x) = 1

sec (x)

periodicity table with

2 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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

General solutions for

sec (x) = - 1

sec (x)

periodicity table with

2 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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = π + 2 πn

x = π + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for -1-2sec^2(x)=-3sec^2(x) ?
The general solution for -1-2sec^2(x)=-3sec^2(x) is x=2pin,x=pi+2pin