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Two rollers of diameters $D_1$ (in mm) and $D_2$ (in mm) are used to measure the internal taper angle in the V-groove of a machined component. The heights $H_1$ (in mm) and $H_2$ (in mm) are measured by using a height gauge after inserting the rollers into the same V-groove as shown in the figure.

Which one of the following is the correct relationship to evaluate the angle $\alpha$ as shown in the figure?

  1. $\sin \alpha = \dfrac{(D_1 – D_2)}{2(H_1-H_2) – (D_1 – D_2)} \\$
  2. $\cos \alpha = \dfrac{(D_1 – D_2)}{2(H_1-H_2) – 2(D_1 – D_2)} \\$
  3. $cosec \: \alpha = \dfrac{(H_1-H_2) – (D_1 – D_2)} {2(D_1 – D_2)}\\$
  4. $\sin \alpha = \dfrac{(H_1-H_2) }{ (D_1 – D_2)} $
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