Let me solve the first equation. The second equation is beyond me.

5*sin X + 7*cos X = 7.830127019 --------(1)

Here is one way.

The first equation involves sinX and cosX. To solve for X, we convert sinX into cosX (or cosX into sinX) so that we have only one variable.

The Pythagorean trig identity:

sin^2(X) +cos^2(X) = 1 -----------(i)

Let us have sinX only, so,

cos^2(X) = 1 -sin^2(X)

cosX = sqrt[1 -sin^2(X)] ------(ii)

Substitute that into (1),

5sinX +7*sqrt[1 -sin^2(X)] = 7.830127019

Isolate the radical,

7*sqrt[1 -sin^2(X)] = 7.830127019 -5sinX

Clear the radical, square both sides,

49[1 -sin^2(X)] = (7.830127019)^2 -2(7.830127019)(5sinX) +25sin^2(X)

49 -49sin^2(X) = 61.31088913 -78.30127019sinX +25sin^2(X)

Bring them all to the rightside, [since the sin^2(X) is positive there. :-)],

0 = [25 +49]sin^2(X) -[78.30127019]sinX +[61.31088913 -49]

0 = 74sin^2(X) -78.30127019sinX + 12.31088913

Get its simplest form, divide both sides by 74,

0 = sin^2(X) -1.058125273sinX +0.166363367

Or,

sin^2(X) -1.058125273sinX +0.166363367 = 0 ----(iii)

That is a quadratic equation in sinX.

Use the quadratic formula and you should get

sinX = 0.866 or 0.1921.

Therefore,

X = arcsin(0.866) or arcsin(0.1921)

X = 60degrees or 11.07536 degrees ----answer.

Or

X = pi/3 radians or 0.1933 radians ----answer.