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TMUA 2018 · Paper 1 · Question 20 of 20

TMUA 2018 Paper 1 Question 20

Trigonometry — Pairing terms with the complementary angle. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 1TrigonometryPairing terms with the complementary angle6 options
Find the value of sin2 0∘ + sin2 1∘ + sin2 2∘ + sin2 3∘ + ⋯ + sin2 87∘ + sin2 88∘ + sin2 89∘ + sin2 90∘
  1. A0.5
  2. B1
  3. C1.5
  4. D45
  5. E45.5
  6. F46
Show the answer and worked solution
answer · E
  1. A0.5
  2. B1
  3. C1.5
  4. D45
  5. E45.5
  6. F46
Pair each term with the one for the complementary angle: sin(90∘ − k∘) = cos k∘, so sin2 k∘ + sin2(90∘ − k∘) = sin2 k∘ + cos2 k∘ = 1. There are 91 terms, from k = 0 to k = 90, which pair up as (0,  90),  (1,  89),  …,  (44,  46) — that is 45 pairs each contributing 1 — leaving the middle term sin2 45∘ = 12 unpaired. The total is 45.5.