use sum of sines and cosines

Question

cos3x-sin3x=0

ok, so I have to find ALL SOLUTIONS to this equation and I get stuck here: 3x =7pi/4 +2kpi how many solutions should there be?????

Answer 1

cos3x - sin3x = 0

Add sin3x to each side.

cos3x = sin3x

Divide each side by cos3x.

tan3x = 1

Trigonometry table in tan45° = 1

So, tan3x = tan45° ⇒ 3x = 45 = π/4

Divide each side by 3.

x = π/12 = (n +1)π/12   where n is some integer.

Comments

tan(3x) = 1.

tan(3x) = tan(π/4).

General solution : If tan θ = tan ∝, then the general solution of θ = nπ + ∝, for n Є Z.

3x = nπ + π/4

⇒ x = (1/3)(nπ + π/4)

x = nπ/3 + π/12.

The general solution is x = π/12(4n + 1) for n Є Z.