Point A lies on a unit circle in the xy-plane and has coordinates \((1, 0)\). Point C is the center...

1. TRANSLATE the problem information

• Given information:
  • Unit circle centered at origin \(\mathrm{C = (0, 0)}\)
  • Point A at \((1, 0)\)
  • Point B at \((x, -1)\) where x is unknown
  • Need positive measure of angle ACB

2. INFER that we need B's exact coordinates first

• Since B lies on the unit circle, it must satisfy the circle equation
• We can solve for the unknown x-coordinate
• Then we'll determine the central angle

3. SIMPLIFY to find point B's coordinates

• Substitute \(\mathrm{B = (x, -1)}\) into unit circle equation \(\mathrm{x^2 + y^2 = 1}\):

\(\mathrm{x^2 + (-1)^2 = 1}\)
\(\mathrm{x^2 + 1 = 1}\)
\(\mathrm{x^2 = 0}\)
\(\mathrm{x = 0}\)

• Therefore \(\mathrm{B = (0, -1)}\)

4. INFER the angle positions in standard form

• Point \(\mathrm{A = (1, 0)}\) corresponds to 0 radians from positive x-axis
• Point \(\mathrm{B = (0, -1)}\) corresponds to \(\mathrm{3\pi/2}\) radians from positive x-axis
• Central angle ACB = \(\mathrm{3\pi/2}\) radians

5. INFER that coterminal angles are valid

• The problem asks for "positive measure," not necessarily the smallest positive measure
• Any angle coterminal with \(\mathrm{3\pi/2}\) works: \(\mathrm{3\pi/2 + 2\pi n}\) where \(\mathrm{n \geq 0}\)

6. SIMPLIFY by testing answer choices

• For coterminal angles, the difference must be exactly a multiple of \(\mathrm{2\pi}\):
• (C) \(\mathrm{19\pi/2 - 3\pi/2 = 16\pi/2 = 8\pi = 4(2\pi)}\) ✓

7. APPLY CONSTRAINTS to select the correct answer

• Only choice (C) differs from \(\mathrm{3\pi/2}\) by an exact multiple of \(\mathrm{2\pi}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students recognize they need to find B's coordinates but don't connect this to the coterminal angle concept. They calculate that angle ACB = \(\mathrm{3\pi/2}\) and then look for answer choices close to this value, not realizing that much larger coterminal angles are equally valid. This leads to confusion since \(\mathrm{3\pi/2 \approx 4.7}\) and none of the answer choices are close to this value. This causes them to get stuck and guess.

Second Most Common Error:

Inadequate APPLY CONSTRAINTS reasoning: Students understand coterminal angles but incorrectly think any multiple of \(\mathrm{\pi}\) works (not just multiples of \(\mathrm{2\pi}\)). They might accept choices like \(\mathrm{9\pi}\) because \(\mathrm{9\pi - 3\pi/2 = 15\pi/2 = 7.5\pi}\), thinking "this is 7.5 times \(\mathrm{\pi}\), so it should work." This may lead them to select Choice B (\(\mathrm{9\pi}\)).

The Bottom Line:

This problem combines coordinate geometry with angle measurement concepts. Students must recognize that finding exact coordinates is the key first step, then apply the coterminal angle concept correctly by checking for differences that are exact multiples of \(\mathrm{2\pi}\).