5.1 The Unit Circle
Ideas in this lesson
- Unit-circle equation and membership
- Recover signed missing coordinate
- Signed real arc and terminal point
- Reference numbers
- Read approximate terminal coordinates
- Transform a known terminal point
- Derive special terminal points
Unit-circle equation and membership · Worked example
A circle consists of points at distance 1 from the origin. State its center, radius and equation. Test \(\left(-1,0\right)\) and \(\left(-1,\dfrac{1}{7}\right)\) by exact squared sums, not decimal estimates. Show the requested reasoning; written work and sketches require teacher review. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: Center \(\left(0,0\right)\); radius \(1\); \(x^2+y^2=1\); first point on, second point off.
\(x^2+y^2=1\) follows from the distance formula with center \(\left(0,0\right)\) and radius \(1\).
\(1+0=1\) verifies the first point.
\(1+\dfrac{1}{49}=\dfrac{50}{49}\), which differs from 1, so the second point is not on the circle.
Written-work review:
\(1+0=1\) verifies the first point.
\(1+\dfrac{1}{49}=\dfrac{50}{49}\), which differs from 1, so the second point is not on the circle.
Written-work review:
- Use the stated definitions and the complete allowed domain.
- Show all intermediate values and justify signs, restrictions and conclusions.
- Give every requested result in the stated order and units.