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TMUA 2019 · Paper 1 · Question 19 of 20

TMUA 2019 Paper 1 Question 19

Trigonometry — Summing a periodic trigonometric series. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 1TrigonometrySumming a periodic trigonometric series6 options
Find the value of ∑k=090 sin(10 + 90k)∘
  1. A0
  2. Bsin 10∘
  3. Csin 100∘
  4. Dsin 190∘
  5. Esin 280∘
  6. F1
Show the answer and worked solution
answer · C
  1. A0
  2. Bsin 10∘
  3. Csin 100∘
  4. Dsin 190∘
  5. Esin 280∘
  6. F1
The terms repeat with period 4 in k: sin 10∘, sin 100∘, sin 190∘ = −sin 10∘, sin 280∘ = −sin 100∘, and each block of four sums to zero. There are 91 terms, and 91 = 4 × 22 + 3, so the 88 terms with k = 0 to 87 cancel completely and only k = 88,  89,  90 survive. Those are sin 10∘ + sin 100∘ + sin 190∘ = sin 10∘ + sin 100∘ − sin 10∘ = sin 100∘.