Expert: Sherry Wallin - 2/11/2011

Question
QUESTION: FIND THE GENERAL SOLUTION OF THE EQUATION COS X COS 6X = −1

ANSWER: Vipul~

What level of math is this from? Specifically, what course is this problem from? Is this from a differential equation course?
One solution is x = pi which can be extended (generalized) to
x = pi + 2n*pi
cos(pi)*cos(2*pi) = -1*1 = -1

Math Prof

---------- FOLLOW-UP ----------

QUESTION: It is a question from trigonometric equation. Please provide the solution by factorizing or any other method of solving trigonometric equation.

Answer
Vipul~

You need to use trigonometric identities to get the angle 6x down to an angle of x. It would also be helpful to get all the terms in cos^a(x) or sin^a(x) where a is some power. However, none of this is necessary since the largest cos u, where 0 < u < 2pi, can be is 1. That is to say you can not get a product of -1 any way other then by 1*-1 or -1*1. The cos u, is at most 1 and at the least -1 and all other values of cos u fall in the interval
-1< cos u <1 and would not give a product of -1 only some value in (-1,1). This means simply that either cos x = 1 or -1 and cos 6x = -1 or 1 respectively in order to get a product of -1. If x was 0 then 6x would be 0 and the product cos 0 * cos(6*0) = 1*1 = 1. If x is pi then 6x = 6pi so cos pi * cos 6pi = -1*1 = -1, thus giving you the product you are searching for.

Math Prof