Plane Trigonometry and Numerical Computation
Chapter VIII
Trigonometric Relations (Continued)
65. Trigonometric Equations. All identity,
as we have seen (§ 26), is an equality between two expressions which is satisfied
for all values of the variables for which both expressions are defined. If the
equality is not satisfied for all values of the variables for which each side is
defined, it is called a conditional equality, or simply an equation. Thus
1 - cos θ = 0 is true only if θ = n·360°,
where n is an integer. To solve a trigonometric equation, i.e. to find the
values of θ for which the equality is true, we usually proceed as follows.
1. Express all the trigonometric functions involved in terms
of one trigonometric function of the same angle.
2. Find the value (or values) of this function by ordinary
algebraic methods.
3. Find the angles between 0° and 360° which correspond to
the values found. These angles are called particular solutions.
4. Give the general solution by adding n·360°, where n is any
integer, to the particular solutions.
EXAMPLE 1. Find θ when sin θ = 1/2.
The particular solutions are 30° and 150°. The general
solutions are 30° + n·360°, 150° + n·360°.
EXAMPLE 2. Solve the equation tan θ sin θ - sin θ = 0.
Factoring the expression, we have sin θ (tan θ - 1) = 0.
Hence we have sin θ = 0, or tan θ - 1 = 0. Why?
The particular solutions are therefore 0°, 180°, 45°, 225°. The
general solutions are n·360°, 180° + n·360°, 45° + n·360°, 225° + n·360°.
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