Identities and Conditional equations
Trigonometric equations can be broken into two categories: identities and
conditional equations. Identities are true for any angle, whereas
conditional equations are true only for certain angles. Identities can be
tested, checked, and created using knowledge of the eight fundamental
identities. We already discussed these processes in Trigonometric Identities
. The following sections are dedicated to
explaining how to solve conditional equations.
Conditional Equations
When solving a conditional equation, a general rule applies: if there is one
solution, then there are an infinite number of solutions. This strange truth
results from the fact that the trigonometric functions are periodic, repeating
every 360 degrees or
2Π
radians. For example, the values of the
trigonometric functions at 10 degrees are the same as they are at 370 degrees
and 730 degrees. The form for any answer to a conditional equation is
θ +2nΠ
, where
θ
is one solution to the equation, and n is an integer. The
shorter and more common way to express the solution to a conditional equation is
to include all the solutions to the equation that fall within the bounds
[0, 2Π)
, and to omit the "
+2nΠ
" part of the solution. since it is assumed
as part of the solution to any trigonometric equation. Because the set of
values from
0
to
2Π
contains the domain for all six trigonometric
functions, if there is no solution to an equation between these bounds, then no
solution exists.
Solutions for trigonometric equations follow no standard procedure, but there
are a number of techniques that may help in finding a solution. These
techniques are essentially the same as those used in solving algebraic
equations, only now we are manipulating trigonometric
functions: we can factor an expression to get different, more understandable
expressions, we can multiply or divide through by a scalar, we can square or
take the square root of both sides of an equation, etc. Also, using the eight
fundamental identities, we can
substitute certain functions for others, or break a functions down into two
different ones, like expressing tangent using sine and cosine. In the problems
below, we'll see just how helpful some of these techniques can be.
problem1
cos(x) =


x = ,


In this problem, we came up with two solutions in the range
[0, 2Π) : x =
, and
x =
. By adding
2nΠ
to either of these
solutions, where
n
is an integer, we could have an infinite number of solutions.
problem2
sin(x) = 2 cos^{2}(x)  1


sin(x) = 2(1  sin^{2}(x))  1


sin(x) = 1  2 sin^{2}(x)


2 sin^{2}(x) + sin(x)  1 = 0


(sin(x) + 1)(2 sin(x)  1) = 0


At this point, after factoring, we have two equations we need to deal with
separately. First, we'll solve
(sin(x) + 1) = 0
, and then we'll solve
(2 sin(x)  1) = 0
problem2a
x =


sin(x) =


x = ,


For the problem, then, we have three solutions:
x = ,,
. All of them check. Here is one more problem.
problem3
sec^{2}(x) + cos^{2}(x) = 2


1 + tan^{2}(x) + 1  sin^{2}(x) = 2


= sin^{2}(x)

