AMC 10 · 2010 · #20

Grade 11 algebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

sequences-geometricpythagorean-identitytrigonometric-ratiosexponents convert-to-algebrapattern-recognition ↑ Prerequisites: sequences-geometricpythagorean-identity

📏 Long solution 💡 3 insights

Problem

Three trigonometric values of one angle form a geometric sequence. Find the position of a given term.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Introduce a Variable

Three trig values forced into one geometric sequence is really one equation in one unknown. Naming c=cos x collapses everything — the ratio, the terms, and the target 1+cos x — into powers of c. Once the sequence is written that way, one term turns out to equal 1, and counting from that term instead of from a₁ makes the whole sequence transparent.

1STEP 1

Turn "geometric" into one equation

Geometric becomes one plain equation.

cos² x=sin xtan x=sin x·(sin x)/(cos x)=(sin² x)/(cos x) ⟹ cos³ x=sin² x

2STEP 2

Bring in sin²x+cos²x=1

The Pythagorean identity closes it.

c³ = 1-c² ⟺ c³+c²-1=0, c=cos x

3STEP 3

Check such an angle really exists

Such an angle really does exist.

f(0)=-1, f(1)=1, c≈ 0.7549, sin² x=c³≈ 0.4302 ＞ 0

4STEP 4

Spot the term that equals 1

The fourth term is exactly 1.

a₄=sin x((cos x)/(sin x))³=(cos³ x)/(sin² x)=(sin² x)/(sin² x)=1

5STEP 5

Count from a₄, not from a₁

Counting from there makes every term a power.

a_n=a₄ r^ n-4=r^ n-4 for all n ≥ 1

6STEP 6

Recognise 1+cos x as a₈

The target matches the term at position 8.

a₈=r⁴=(cos⁴ x)/(sin⁴ x)=(cos⁴ x)/(cos⁶ x)=1/(cos² x), c²(1+c)=1 ⟹ 1+cos x=1/(cos² x)

7STEP 7

No other index can work

Strict growth means no other index works.

|r|=1/√(c)≈ 1.151 ＞ 1, |a_n|=|r|^ n-4 strictly increasing ⟹ n=8

Answer

Put numbers on it. The cubic gives cos x≈ 0.754878, so sin x≈ 0.655866 and tan x≈ 0.868837; check that 0.655866/0.754878≈ 0.868837, so the first three terms really are geometric with r≈ 1.150964. The sequence runs 0.6559, 0.7549, 0.8688, 1, 1.1510, 1.3247, 1.5247, 1.7549, and 1+cos x≈ 1.754878 — matching a₈ and nothing before it. The other candidate indices give 1, 1.1510, 1.3247, 1.5247, all clearly too small. One case worth noting: cos x is forced, but sin x may be negative (x≈-0.7153 instead of x≈ 0.7153). That flips the sign of r and of the odd-offset terms a₅,a₇, yet a₄=1 and a₈=r⁴≈ 1.754878 are unchanged, so the answer does not depend on which of the two angles is meant.

💡Key takeaway

When one term of a geometric sequence turns out to be 1, count from there — every other term is just the ratio raised to how far away it sits.

Turn "geometric" into one equation
Bring in sin²x+cos²x=1
Check such an angle really exists
Spot the term that equals 1
Count from a₄, not from a₁
Recognise 1+cos x as a₈
No other index can work