Question

Solve 4sin(x)+3cos(x) = 2 for the first 2 positive solutions

Answer 1

Refer to the attached solution using Mathematica 9.

Answer 2

Divide by cos
\[4 \tan x + 3 = 2 \sec x\]
square both sides
\[16 \tan^2 x + 24 \tan x + 9 = 4 \sec^2 x\]
use identity: sec^2 = 1+tan^2
\[16 \tan^2 x + 24 \tan x + 9 = 4 + 4 \tan^2 x\]
\[12 \tan^2 x + 24 \tan x + 5 = 0\]
\[\tan x = \frac{-24 \pm \sqrt{24^2 - 4(12)(5)}}{2(12)}\]
simplified
\[\tan x = -1 \pm \frac{\sqrt{21}}{6}\]
\[x = \pi n + \tan^{-1} ( -1 \pm \frac{\sqrt{21}}{6})\]
From here use a calculator to estimate the first 2 positive solutions

Answer 3

sorry I forgot to add you need to also check for extraneous solutions
the above solution gives the following positive solutions
\[x \approx 2.08, 2.91, 5.22, 6.05\]
two are extraneous, dont work if you plug into original equation
real solutions:
\[x \approx 2.08, 6.05\]

Answer 4

@dumbcow
Nice work.

Mathematica allows someone like myself, 80 years old this August, to particpate on this site.

I'm not sure what the derivative of x^2 is. At this time I cannot remember any of the derivative rules. Somewhere in that 1.5 Gb installation file, Mathematica has access to all of them.