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Q: What are small circles according to algebra and trigonometry?

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You need some more information about any of the circles, or about the relationship between the two circles, to answer this question.

yes because one is smaller than the other

Historically, it is because we live on a planet which is approximately spherical. 2-dimensional trigonometry was adequate for relatively small shapes where the curvature of the earth had negligible effect. For larger shapes the spherical nature of the earth was important and therefore, so was spherical trigonometry.

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