ARCH-MM-06: Radian Sector Mensuration & Counting Trigonometric Roots in Transformed Intervals

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-MM-06: Why the 160s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~160s (Blows 90s Clock)

Listing every individual angle

u = 3x

[0, 6\pi]

and dividing each by

3

to list all

x

-values takes ~130s and yields

6

solutions.

The Fast ShortcutTarget: 45s (Saves 115s)

Line 1 (Interval Scaling): Let

u = 3x \in [0, 6\pi]

, which is

3

full wave cycles of length

2\pi

.
Line 2 (Cycle Multiplication): Each cycle contributes

2

solution(s) plus

0

right-endpoint hit

= 3(2) + 0 = 6

(25s).

Where Students Lose Time: In 53 past UAT-UK questions (12.93% of Mathematics 2), candidates lose 115s by solving for theta in one period and dividing by k instead of scaling the interval to u = ktheta first. Instead, apply MM4 with Maths 1 skill M5.8*.

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