Question

Sin6x-cos6x= class="latex-formula">

Answer 1

Condense the right side a single sine expression:

sin(6x) - cos(6x) = R sin(6x - t)

Expanding the right side gives

sin(6x) - cos(6x) = R sin(6x) cos(t) - R cos(6x) sin(t)

Then we have

R cos(t) = 1

R sin(t) = 1

Solve for R and t:

(R cos(t))² + (R sin(t))² = 1² + 1²

R² = 2

R = √2

and

(R sin(t))/(R cos(t)) = 1/1

tan(t) = 1

t = arctan(1) = π/4

So we rewrite the equation as

√2 sin(6x - π/4) = √2

Solve for x :

sin(6x - π/4) = 1

6x - π/4 = arcsin(1) + 2nπ

(where n is any integer)

6x - π/4 = π/2 + 2nπ

6x = 3π/4 + 2nπ

x = π/8 + nπ/3