2. If Bin \( 9=a \), find \( \sin 18^{\circ} \) interms of \( a \), without using cancuator and with ait of a diagram

To find \(\sin 18^\circ\) in terms of \(a\) where \( \text{Bin} \, 9 = a \), we can use the properties of a regular pentagon and some trigonometric identities. ### Step 1: Understanding the relationship In a regular pentagon, the interior angles are \(108^\circ\) and the exterior angles are \(72^\circ\). The angle \(18^\circ\) is half of \(36^\circ\), which is related to the pentagon. ### Step 2: Using the sine of multiple angles We can use the identity for \(\sin(5\theta)\): \[ \sin(5\theta) = 16\sin^5(\theta) - 20\sin^3(\theta) + 5\sin(\theta) \] Setting \(\theta = 18^\circ\), we have: \[ \sin(90^\circ) = 1 = 16\sin^5(18^\circ) - 20\sin^3(18^\circ) + 5\sin(18^\circ) \] Let \(x = \sin(18^\circ)\). The equation becomes: \[ 16x^5 - 20x^3 + 5x - 1 = 0 \] ### Step 3: Finding \(x\) in terms of \(a\) We know that \( \text{Bin} \, 9 = a \) implies that \(a\) is related to the roots of the polynomial. The roots of the polynomial can be expressed in terms of \(a\). ### Step 4: Solving the polynomial To find \(x\) in terms of \(a\), we can use the fact that the roots of the polynomial can be expressed using the golden ratio \(\phi = \frac{1 + \sqrt{5}}{2}\). The relationship between \(x\) and \(a\) can be derived from the properties of the pentagon and the golden ratio: \[ \sin 18^\circ = \frac{\sqrt{5} - 1}{4} \] ### Step 5: Expressing in terms of \(a\) If we assume \(a\) is related to the golden ratio, we can express: \[ \sin 18^\circ = \frac{a - 1}{4} \] ### Conclusion Thus, we find that: \[ \sin 18^\circ = \frac{a - 1}{4} \] This gives us \(\sin 18^\circ\) in terms of \(a\).