solve trigonometric equation

Question

cosx cos2x+sinx sin2x=1/2

Answer 1

Given Cosx Cos2x+Sinx Sin2x = 1/2

Cos2x Cosx+Sin2x Siinx = 1/2

Remember the formula Cos(A-B) = CosA CosB+SinA SinB

Cos(2x-x) = 1/2

Cosx = Cos 60

x = 60

Comments

cos(x) = 1/2.

x = cos^{- 1}(1/2).

The function cos (x) has a period of 2π, first find all solutions in the interval (0, 2π).

The function cos (x) is positive in first and fourth quadrant.

• cos(x) = 1/2.

In first Quadrant, 0 ≤ x ≤ π/2.

1/2 = cos (π/3) = cos (0 + π/3) = cos (π/3).

In fourth Quadrant, 3π/2 ≤ x ≤ 2π.

1/2 = cos (π/3) = cos (2π - π/3) = cos (5π/3).

So, the general solutions are x = 2n₁π + π/3 and x = 2n₂π + 5π/3, where n₁, n₂∈ Z.