Trig equations and general solutions

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What this lesson covers

  • sin x = sin α: x = nπ + (−1)ⁿα.
  • cos x = cos α: x = 2nπ ± α.
  • tan x = tan α: x = nπ + α.
Key idea

The three forms

  • In each one, α is a known solution and n is any integer.
  • Solve for the whole bracket first when the equation has 2x or 3x.
Worked example

sin x = 1/2

  • The known angle is π/6.
  1. α = π/6
  2. x = nπ + (−1)ⁿ π/6
Worked example

2cos²x − cos x − 1 = 0

  • It is a quadratic in cos x.
  1. (2cos x + 1)(cos x − 1) = 0
  2. cos x = 1 gives x = 2nπ
  3. cos x = −1/2 gives x = 2nπ ± 2π/3
Worked example

cos 2x = 1/2, for 0 ≤ x < 2π

  • Solve for 2x, then divide.
  1. 2x = 2nπ ± π/3
  2. x = nπ ± π/6
  3. x = π/6, 5π/6, 7π/6, 11π/6

Common mistakes

  • Dividing by cos x in an equation.
    That loses solutions, so factorise instead.
  • Giving one answer with no n.
    A general solution needs n.

Your turn

  • 1. Solve tan x = 1 in general.
  • 2. Solve 2 sin x = √3 in general.
Show answer

1. x = nπ + π/4 2. x = nπ + (−1)ⁿ π/3

Summary

What to take away

  • Solve for the bracket first.
  • Factorise, never divide by a trig function.

More in AS 91575

Trig functions and radiansNotesTrig identities and proofsNotesCompound and double-angle formulaeNotesTrig equations and general solutionsNotesModelling periodic situationsNotes

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