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TMUA 2016 · Paper 1 · Question 10 of 20

TMUA 2016 Paper 1 Question 10

Trigonometry — Counting solutions · intersecting tan x with a reciprocal. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 1TrigonometryCounting solutions · intersecting tan x with a reciprocal7 options
How many solutions does the equation xtan x = 1 have in the interval −2π ≤ x ≤ 2π?
  1. A0
  2. B1
  3. C2
  4. D3
  5. E4
  6. F5
  7. G6
Show the answer and worked solution
answer · E
  1. A0
  2. B1
  3. C2
  4. D3
  5. E4
  6. F5
  7. G6
Rewrite it as tan x = 1x (note x = 0 is not a solution) and count crossings of y = tan x with the hyperbola y = 1x. The function xtan x is even, so it is enough to count on (0,  2π] and double. On (0,  π2), tan x climbs from 0 past the falling 1x, giving one crossing; on (π2,  3π2) it climbs from −∞ to +∞ while 1x stays small and positive, giving one more; on (3π2,  2π] tan x is negative while 1x is positive, so there are none. That is 2 on the positive side and 2 on the negative side, so 4 in total.