# 4.6
Solve[x2^2 + (y2-L4)^2 == L3^2 && (x2-L1 Cos[q1])^2+(y2-L2 Sin[q1])^2 == L2^2, {x2, y2}]

# 4.7
x = {{0, -1, 0, 0, 0},  {Cos[q2], 0, 0, 0, 0}, { Sin[q2], 0, 1, 0, 0}, {0, 0, R*Cos[q1], 1, 0}, {0, 0, R*Sin[q1], 0, 1}};
x // MatrixForm
Inverse[x]