If \( \sin 9^{\circ}=\alpha \) Find \( \sin 18^{\circ} \) in terms of a without using a colculator and with the aid of a diagram
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To find \( \sin 18^{\circ} \) in terms of \( \alpha \), we can utilize the double angle formula for sine. First, we know: \[ \sin 18^{\circ} = \sin(2 \times 9^{\circ}) = 2 \sin 9^{\circ} \cos 9^{\circ} \] Since \( \sin 9^{\circ} = \alpha \), we need to express \( \cos 9^{\circ} \) in terms of \( \alpha \). We use the Pythagorean identity: \[ \cos^2 9^{\circ} = 1 - \sin^2 9^{\circ} = 1 - \alpha^2 \] Taking the positive square root (since \( 9^{\circ} \) is in the first quadrant): \[ \cos 9^{\circ} = \sqrt{1 - \alpha^2} \] Now, substituting back into the double angle formula for sine: \[ \sin 18^{\circ} = 2 \sin 9^{\circ} \cos 9^{\circ} = 2 \alpha \sqrt{1 - \alpha^2} \] So, \( \sin 18^{\circ} \) can be expressed as \( 2\alpha \sqrt{1 - \alpha^2} \). --- To illustrate this further, imagine a right triangle where the angle is \( 18^{\circ} \) and the opposite side represents \( \sin 18^{\circ} \) while the adjacent side is related to \( \cos 9^{\circ} \). The triangle will help visualize how these angle relationships work collectively in determining the values of sine and cosine. Lastly, for context, \( \sin 18^{\circ} \) has a fascinating geometric interpretation related to the pentagon and golden ratio, deepening our understanding of sine functions in polygons and angles.