The length of the chord joining the points $\left( {4\cos \theta ,4\sin \theta } \right)$ and $\left[ {4\cos \left( {\theta + {{60}^o}} \right),4\sin \left( {\theta + {{60}^o}} \right)} \right]$ of the circle ${x^2} + {y^2} = 16$ is:

Both given points lie on the circle x^2 + y^2 = 16, meaning they are radius vectors of length 4 originating from the origin. The angle between these two radius vectors is 60 degrees. Using the distance formula or the cosine rule in the triangle formed by the two radii and the chord, the length of the chord is sqrt(4^2 + 4^2 - 2 * 4 * 4 * cos(60 degrees)). This simplifies to sqrt(32 - 16) = sqrt(16), giving a chord length of 4.

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