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Popular >

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

Solution

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

Solution

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = sin^{2} (x)

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

Multiply the numbers:

2 \cdot 1 = 2

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

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

Simplify

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

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

Group like terms

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

- 2 + 2 = 0

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

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

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

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

Factor

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

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

Rewrite

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

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

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

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

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

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (sin (2 x) + 2 sin (x)) (sin (2 x) - 2 sin (x))

(sin (2 x) + sin (x) 2) (sin (2 x) - sin (x) 2) = 0

Solving each part separately

sin (2 x) + sin (x) 2 = 0 or sin (2 x) - sin (x) 2 = 0

sin (2 x) + sin (x) 2 = 0 : x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sin (2 x) + sin (x) 2 = 0

Rewrite using trig identities

sin (2 x) + sin (x) 2

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (x) cos (x) + 2 sin (x)

sin (x) 2 + 2 cos (x) sin (x) = 0

Factor

sin (x) 2 + 2 cos (x) sin (x) : sin (x) (2 + 2 cos (x))

sin (x) 2 + 2 cos (x) sin (x)

Factor out common term

sin (x)

= sin (x) (2 + 2 cos (x))

sin (x) (2 + 2 cos (x)) = 0

Solving each part separately

sin (x) = 0 or 2 + 2 cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

2 + 2 cos (x) = 0 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

2 + 2 cos (x) = 0

Move

2

to the right side

2 + 2 cos (x) = 0

Subtract

2

from both sides

2 + 2 cos (x) - 2 = 0 - 2

Simplify

2 cos (x) = - 2

2 cos (x) = - 2

Divide both sides by

2

2 cos (x) = - 2

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{- 2}{2}

Simplify

cos (x) = - \frac{2}{2}

cos (x) = - \frac{2}{2}

General solutions for

cos (x) = - \frac{2}{2}

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

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sin (2 x) - sin (x) 2 = 0 : x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

Rewrite using trig identities

sin (2 x) - sin (x) 2

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

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

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

Factor

- sin (x) 2 + 2 cos (x) sin (x) : sin (x) (- 2 + 2 cos (x))

- sin (x) 2 + 2 cos (x) sin (x)

Factor out common term

sin (x)

= sin (x) (- 2 + 2 cos (x))

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

Solving each part separately

sin (x) = 0 or - 2 + 2 cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

- 2 + 2 cos (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

- 2 + 2 cos (x) = 0

Move

2

to the right side

- 2 + 2 cos (x) = 0

Add

2

to both sides

- 2 + 2 cos (x) + 2 = 0 + 2

Simplify

2 cos (x) = 2

2 cos (x) = 2

Divide both sides by

2

2 cos (x) = 2

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{2}{2}

Simplify

cos (x) = \frac{2}{2}

cos (x) = \frac{2}{2}

General solutions for

cos (x) = \frac{2}{2}

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

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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