2. If Bin \( 9=a \), find \( \sin 18^{\circ} \) interms of \( a \), without using cancuator and with ait of a diagram
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To find \( \sin 18^{\circ} \) in terms of \( a \) (where \( a = \binom{9}{2} \)), we first recognize that \( a = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \). From trigonometric identities, we can use the fact that \( \sin(18^{\circ}) = \frac{\sqrt{5} - 1}{4} \). To express this in terms of \( a \), we set up the equation \( 36 = 4(\sqrt{5} - 1) \). Thus, \( \sin 18^{\circ} = \frac{a}{9} + \frac{1}{4} \). Check if your diagram includes a pentagon, as this could help visualize \( 18^{\circ} \) as an internal angle. The golden ratio also plays a role here since \( \sin 18^{\circ} \) relates to the properties of a regular pentagon! Drawing lines connecting the vertices can make the relationship clearer and show how \( \sin 18^{\circ} \) appears in geometrical figures.