2sec xsin x + 2 = 4sin x + sec x2secxsinx+2=4sinx+secx
(2sin x)/(cos x) + 2 = (4sinxcos x + 1)/(cos x)2sinxcosx+2=4sinxcosx+1cosx
2(sin x + cos x) = 4sinxcos x + 12(sinx+cosx)=4sinxcosx+1 (1)
- Multiplying both side by cos x (condition cos x diff. to zero)
Call (sin x + cos x ) = u(sinx+cosx)=u
u^2 = (sin x + cos x)^2 = sin^2 x + cos^2 x + 2sinxcos x u2=(sinx+cosx)2=sin2x+cos2x+2sinxcosx
= 1 + 2sin xcos x=1+2sinxcosx.
2sin xcos x = u^2 - 12sinxcosx=u2−1
4sin xcos x = 2u^2 - 24sinxcosx=2u2−2
Substitute these values into (1):
2u = 2u^2 - 2 + 12u=2u2−2+1
2u^2 - 2u - 1 = 02u2−2u−1=0
Solve this quadratic equation for u = (sin x +cos x)u=(sinx+cosx)
D = d^2 = b^2 - 4ac = 4 + 8 = 12D=d2=b2−4ac=4+8=12 --> d = +- 2sqrt3d=±2√3
There are 2 real roots:
sin x + cos x = u = 2/2 +- (2sqrt3)/4 = 2 +- sqrt3/2sinx+cosx=u=22±2√34=2±√32
u_1 = 1 + sqrt3/2u1=1+√32 (Rejected as >sqrt2>√2)
u_2 = 1 - sqrt3/2 = 0.134u2=1−√32=0.134
sin x + cos x = sqrt2cos (x - pi/4) = 0.134sinx+cosx=√2cos(x−π4)=0.134
cos (x - pi/4) = 0.134/sqrt2 = 0.0947cos(x−π4)=0.134√2=0.0947
x - pi/4 = +- 84.564^@x−π4=±84.564∘
x = 84.564 + 45 = 129.564^@x=84.564+45=129.564∘ and
x = 360- 84.564 + 45 = 321.564^@x=360−84.564+45=321.564∘
