3.3: Solving Trig Equations using Identities

Question

Find all solutions of the equation 

The answer is  where  is any integer,the constant

 

Answer 1

tan5x - 9tanx = 0; tan5x - tanx = 8tanx; sin4x/(cos5xcosx) = 8sinx/cosx; sin4x = 8sinxcos5x;

sin4x = 4(sin6x - sin4x); 5sin4x = 4sin6x; 10sin2xcos2x = 4sin2x(3-4sin²2x); 5sin2xcos2x = 2sin2x(3 -

4sin²2x); sin2x = 0; 2x = πn; x = πn/2; 5cos2x = 6 - 8+ 8cos²2x; 8cos²2x - 5cos2x - 2 = 0;

cos2x = (5 ± √25 + 64)/16; cos2x = (5 ± √89)/16;

x = 1/2cos^-1[(5 ± √89)/16] + πn; x = π -1/2cos^-1[(5 ±√89)/16} +πn