The cosine function has positive values in the first and fourth quadrants, and negative values in the second and third. The value you found is in the second quadrant: π/2 < 1.97 < π
To find the corresponding angle in the third quadrant, you first need to find the reference angle.
π - φ = 1.97
φ = π - 1.97
This is the measure of the angle between the -x-axis and the angle you've found. To put this into the third quadrant, you need to add it to the -x-axis
π + φ = the 2nd answer.
In general, when you have an angle and want to find similar angles in other quadrants, you'll use the following relationships between standard angle θ and reference angle φ. All 4 of these angles will have the same absolute value of each trig function. You'll need to learn which trig functions are positive and negative in which quadrants.
In QI: θ = φ
QII: θ = π - φ
QIII: θ = π + φ
QIV: θ = 2π - φ,
or θ = -φ for negative angles -π/2 < θ < 0