What is the general solution for 2sin^2(x)-cos^2(x)-2=0 ?

The general solution for 2sin^2(x)-cos^2(x)-2=0 is x= pi/2+2pin,x=(3pi)/2+2pin

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2sin^2(x)-cos^2(x)-2=0

Solution

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

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

- 2 - cos^{2} (x) + 2 sin^{2} (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= - 2 - (1 - sin^{2} (x)) + 2 sin^{2} (x)

Simplify

- 2 - (1 - sin^{2} (x)) + 2 sin^{2} (x) : 3 sin^{2} (x) - 3

- 2 - (1 - sin^{2} (x)) + 2 sin^{2} (x)

- (1 - sin^{2} (x)) : - 1 + sin^{2} (x)

- (1 - sin^{2} (x))

Distribute parentheses

= - (1) - (- sin^{2} (x))

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 1 + sin^{2} (x)

= - 2 - 1 + sin^{2} (x) + 2 sin^{2} (x)

Simplify

- 2 - 1 + sin^{2} (x) + 2 sin^{2} (x) : 3 sin^{2} (x) - 3

- 2 - 1 + sin^{2} (x) + 2 sin^{2} (x)

Add similar elements:

sin^{2} (x) + 2 sin^{2} (x) = 3 sin^{2} (x)

= - 2 - 1 + 3 sin^{2} (x)

Subtract the numbers:

- 2 - 1 = - 3

= 3 sin^{2} (x) - 3

= 3 sin^{2} (x) - 3

= 3 sin^{2} (x) - 3

- 3 + 3 sin^{2} (x) = 0

Solve by substitution

- 3 + 3 sin^{2} (x) = 0

Let:

sin (x) = u

- 3 + 3 u^{2} = 0

- 3 + 3 u^{2} = 0 : u = 1, u = - 1

- 3 + 3 u^{2} = 0

Move

3

to the right side

- 3 + 3 u^{2} = 0

Add

3

to both sides

- 3 + 3 u^{2} + 3 = 0 + 3

Simplify

3 u^{2} = 3

3 u^{2} = 3

Divide both sides by

3

3 u^{2} = 3

Divide both sides by

3

\frac{3 u ^{2}}{3} = \frac{3}{3}

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 = sin (x)

sin (x) = 1, sin (x) = - 1

sin (x) = 1, sin (x) = - 1

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

General solutions for

sin (x) = 1

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

General solutions for

sin (x) = - 1

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(x)-cos^2(x)-2=0 ?
The general solution for 2sin^2(x)-cos^2(x)-2=0 is x= pi/2+2pin,x=(3pi)/2+2pin