The blue vector is the sum of the 5 and 12 N magnitude vectors ... the red vector is the opposite of that sum.
Let the origin be at the initial point of both the 5 and 12 N vectors and let the 12 N vector lie on the positive x-axis
$(5\cos{\theta} + 12)^2 + (5\sin{\theta})^2 = 15^2$
$25\cos^2{\theta} + 120 \cos{\theta} + 144 + 25\sin^2{\theta} = 225$
$120\cos{\theta} = 225-144-25$
$\theta = \arccos\left(\dfrac{225-144-25}{120}\right) \approx 62^\circ$