Question #abd69

2 Answers
Jun 21, 2016

(72^@26 + k180)(7226+k180)
(220^@ + k180)(220+k180)

Explanation:

Apply these identities, calling tan (x/2) = ttan(x2)=t
sin x = (2t)/(1 + t^2) sinx=2t1+t2
cos x = (1 - t^2)/(1 + t^2)cosx=1t21+t2
2sin x - 3cos x = 1
Replace sin x and cos x by the above identities, we get:
4t - 3 + 3t^2 = 1 + t^24t3+3t2=1+t2
2t^2 + 4t - 4 = 02t2+4t4=0
t^2 + 2t - 2 = 0t2+2t2=0.
Solve this quadratic equation for t, using the improved quadratic formula (Google, Yahoo Search);
D = d^2 = b^2 - 4ac = 4 + 8 = 12D=d2=b24ac=4+8=12 --> d = +- 2sqrt3d=±23
There are 2 real roots:
t = -b/(2a) +- d/(2a) = -2/2 +- (2sqrt3)/2 = -1 +- sqrt3t=b2a±d2a=22±232=1±3
t1 = -1 -sqrt3 = - 2.73t1=13=2.73
t2 = -1 + sqrt3 = 0.73t2=1+3=0.73
a. t1 = tan (x/2) = -2.73t1=tan(x2)=2.73 --> x/2 = - 69^@88x2=6988 and x/2 = -69.88 + 180 = 110^@12x2=69.88+180=11012 --> x = 220.24 + k180
b. t2 = 0.73 --> x/2 = 36.13x2=36.13 and x/2 = 36.13 + 180 = 216.13x2=36.13+180=216.13 -->
x = 72.26 + k180x=72.26+k180

Jun 21, 2016

Using identities of fractional angle

sinx=(2tan(x/2))/(1+tan^2(x/2))sinx=2tan(x2)1+tan2(x2)

and
cosx=(1-tan^2(x/2))/(1+tan^2(x/2))cosx=1tan2(x2)1+tan2(x2)

Given equation becomes

2sinx-3cosx=12sinx3cosx=1

=(4tan(x/2))/(1+tan^2(x/2))-3((1-tan^2(x/2)))/(1+tan^2(x/2))=1=4tan(x2)1+tan2(x2)3(1tan2(x2))1+tan2(x2)=1

putting tan(x/2)=ttan(x2)=t we get

=>(4t)/(1+t^2)-(3(1-t^2))/(1+t^2)=14t1+t23(1t2)1+t2=1

=>4t-3+3t^2=1+t^24t3+3t2=1+t2

=>3t^2-t^2+4t-3-1=03t2t2+4t31=0

=>2t^2+4t-4=02t2+4t4=0

=>t^2+2t-2=0t2+2t2=0

=>t^2+2t+1=3t2+2t+1=3

=>(t+1)^2=3(t+1)2=3

=>t+1=+-sqrt3t+1=±3

=>t=+-sqrt3-1t=±31

when t =sqrt3-1t=31

tan(x/2)=sqrt3-1tan(x2)=31

:.x=2xxtan^-1(sqrt3-1)=2*36.2^@=72.4^@

Or,x=(180+72.4)=252.4^@

Again when

tan(x/2)=-sqrt3-1

=>x=2xxtan^-1(-sqrt3-1)

=>x=2*110.1^@=220.2^@