How do to derive the following SIMPLE geometric relationship between two points on a plane

Question

Can someone show why:

$$x' = L_1 \cos(a_1) + L_2\cos(a_1+a_2)$$

$$y' = L_1 \sin(a_1) + L_2\sin(a_1+a_2)$$

where $L_1$ and $L_2$ are the length of the red lines

Answer 1

In polar coordinates, if the angle is $a_1$ and the radius is $L_1$, then the Cartesian coordinates of the point are $\displaystyle\left[\begin{array}{l} L_1\cos a_1 \\ L_1\sin a_1 \end{array}\right]$.

If the angle is $a_1+a_2$ and the radius is $L_2$, the you get $\displaystyle\left[\begin{array}{l} L_2\cos (a_1+a_2) \\ L_2\sin (a_1+a_2) \end{array}\right]$.

They're just adding those two vectors as follows. Go from the origin to the coordinates of the first vector; then add the second vector to it by moving from there a distance and in a direction given by the second vector.