How to find the general solution 5 sin(x)+2 cos(x)=3?

2 Answers
May 10, 2018

#rarrx=npi+(-1)^n*(sin^(-1)(3/sqrt29))-sin^(-1)(2/sqrt29)# #n inZZ#

Explanation:

#rarr5sinx+2cosx=3#

#rarr(5sinx+2cosx)/(sqrt(5^2+2^2))=3/(sqrt(5^2+2^2)#

#rarrsinx*(5/sqrt(29))+cosx*(2/sqrt(29))=3/sqrt29#

Let #cosalpha=5/sqrt29# then #sinalpha=sqrt(1-cos^2alpha)=sqrt(1-(5/sqrt29)^2)=2/sqrt29#

Also, #alpha=cos^(-1)(5/sqrt29)=sin^(-1)(2/sqrt29)#

Now, given equation transforms to

#rarrsinx*cosalpha+cosx*sinalpha=3/sqrt29#

#rarrsin(x+alpha)=sin(sin^(-1)(3/sqrt29))#

#rarrx+sin^(-1)(2/sqrt29)=npi+(-1)^n*(sin^(-1)(3/sqrt29))#

#rarrx=npi+(-1)^n*(sin^(-1)(3/sqrt29))-sin^(-1)(2/sqrt29)# #n inZZ#

May 11, 2018

#x = 12^@12 + k360^@#
#x = 124^@28 + k360^@#

Explanation:

5sin x + 2cos x = 3.
Divide both sides by 5.
#sin x + 2/5 cos x = 3/5 = 0.6# (1)
Call #tan t = sin t/(cos t) = 2/5# --> #t = 21^@80# --> cos t = 0.93.
The equation (1) becomes:
#sin x.cos t + sin t.cos x = 0.6(0.93)#
#sin (x + t) = sin (x + 21.80) = 0.56#
Calculator and unit circle give 2 solutions for (x + t) -->
a. x + 21.80 = 33.92
#x = 33.92 - 21.80 = 12^@12#
b. x + 21.80 = 180 - 33.92 = 146.08
#x = 146.08 - 21.80 = 124^@28#
General answers:
#x = 12^@12 + k360^@#
#x = 124^@28 + k360^@#
Check by calculator.
#x = 12^@12# --> 5sin x = 1.05 --> 2cos x = 1.95
5sin x + 2cos x = 1.05 + 1.95 = 3. Proved.
#x = 124^@28# --> 5sin x = 4.13 --> 2cos x = -1.13
5sin x + 2cos x = 4.13 - 1.13 = 3. Proved.