Determine the points on the unit circle corresponding to multiples of $ frac{π}{4} $ and explain their significance in a unit circle art project

To determine the points on the unit circle for multiples of $ \frac{π}{4} $, we first note that:

$ \theta = n \cdot \frac{π}{4} $

where $ n $ is an integer. Evaluating this for $ n = 0, 1, 2, 3, 4, 5, 6, 7 $, we get the following points on the unit circle:

– For $ n = 0 $: $ (\cos(0), \sin(0)) = (1, 0) $

– For $ n = 1 $: $ (\cos(\frac{π}{4}), \sin(\frac{π}{4})) = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) $

– For $ n = 2 $: $ (\cos(\frac{π}{2}), \sin(\frac{π}{2})) = (0, 1) $

– For $ n = 3 $: $ (\cos(\frac{3π}{4}), \sin(\frac{3π}{4})) = (-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) $

– For $ n = 4 $: $ (\cos(π), \sin(π)) = (-1, 0) $

– For $ n = 5 $: $ (\cos(\frac{5π}{4}), \sin(\frac{5π}{4})) = (-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) $

– For $ n = 6 $: $ (\cos(\frac{3π}{2}), \sin(\frac{3π}{2})) = (0, -1) $

– For $ n = 7 $: $ (\cos(\frac{7π}{4}), \sin(\frac{7π}{4})) = (\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) $

These points are significant in a unit circle art project as they help in creating symmetrical designs and patterns based on rotational symmetry.

To find the points on the unit circle at multiples of $ frac{π}{4} $, calculate:

$ heta = n cdot frac{π}{4} $

For $ n = 0 $ to $ n = 7 $, points are:

• $(1, 0)$
• $(frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$
• $(0, 1)$
• $(-frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$
• $(-1, 0)$
• $(-frac{sqrt{2}}{2}, -frac{sqrt{2}}{2})$
• $(0, -1)$
• $(frac{sqrt{2}}{2}, -frac{sqrt{2}}{2})$

These points form an octagon shape, useful in creating designs.

Points for multiples of $ frac{π}{4} $ are:

• $(1, 0)$
• $(frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$
• $(0, 1)$
• $(-frac{sqrt{2}}{2}, frac{sqrt{2}}{2

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