AMC 10 · 2020 · #9
Grade 11 algebraPick an answer.
There is no algebraic trick that turns tan(2x) = cos(x/2) into a solvable equation — the two sides have different periods and different shapes. So Tool #1 (Draw a Diagram): sketch y = tan(2x) and y = cos(x/2) on the same axes and count crossings. Tool #7 (Identify Subproblems): the asymptotes of tan(2x) chop [0, 2π] into separate branches, and each branch is its own small counting problem. Tool #14 (Extreme Principle): on each branch, only the behaviour at the two ends matters — one side runs to -∞ or the other to +∞, which forces a crossing. Tool #3 (Eliminate Possibilities): add the per-branch counts and match the single choice that fits.
Turn the equation into two graphs
Read it as two graphs meeting.
An equation with two hard sides becomes a picture with two curves, and crossings are easy to count.
11.A-REI.D.11Draw A DiagramLocate the asymptotes of tan(2x)
Locate the tangent's asymptotes.
Doubling the input halves the period, so four asymptotes fit where the plain tangent would have only two.
Doubling the input halves the period, so twice as many repeats fit into the same window.
▸ Why?
The pattern returns to its start after one full period, so the period is what sets the spacing.
▸ Why?
A period is a fixed share of the full turn, so scaling the angle scales that share directly.
Track the cosine across the window
The cosine moves gently across the window.
Halving the input doubles the period, so on this window the cosine only has time to fall once — no wiggles.
11.F-BF.B.3Draw A DiagramSplit the window into five branches
The window splits into five branches.
Rising curve minus falling curve can only cross zero going up, and it only gets one chance per branch.
9.F-IF.B.4Identify SubproblemsThe three middle branches each cross once
The middle three each cross once.
A curve sweeping the whole vertical range must pass through a band that only reaches from -1 to 1.
9.F-IF.B.4Extreme PrincipleCheck the first branch from the ends
Check the first branch at its ends.
The tangent starts below the cosine and ends above it, so somewhere in between they were equal.
11.F-TF.A.2Extreme PrincipleCheck the last branch from the ends
Check the last branch too.
The cosine has bottomed out at -1 while the tangent has climbed back to 0, so the tangent overtakes it before the window closes.
11.F-TF.A.2Extreme PrincipleAdd the branches and pick the choice
Add the branches.
Five branches, one crossing apiece — the count is just the number of pieces the asymptotes made.
11.A-REI.D.11Eliminate PossibilitiesSanity-check with sample values
Sample values confirm 5.
A sign change on a continuous piece is hard evidence that a crossing really lives there.
9.F-IF.B.4Eliminate PossibilitiesDo not try to solve tan(2x) = cos(x/2) — draw it. The asymptotes of tan(2x) sit at π/4, 3π/4, 5π/4, 7π/4, cutting [0, 2π] into 5 branches, and on each branch a rising tangent meets the falling, boxed-in cos(x/2) exactly once. Count the branches and you have counted the answer: 5, choice (E).
- Turn the equation into two graphs
- Locate the asymptotes of tan(2x)
- Track the cosine across the window
- Split the window into five branches
- The three middle branches each cross once
- Check the first branch from the ends
- Check the last branch from the ends
- Add the branches and pick the choice
- Sanity-check with sample values