Answer:
(b) 0 . . . . a possible extraneous solution
(c) pi . . . . a possible extraneous solution
There are no actual extraneous solutions.
Explanation:
You want the extraneous solutions for the equation cos(x)cot(x) = 3 -3sin(x) on the interval [0, 2π).
Extraneous solutions
An extraneous solution is a solution of a transformed equation that is not a solution of the original equation. Extraneous solutions can arise when equations are raised to an even power, or are multiplied by factors that are 0 or ∞ for some values of the variable(s). In general, extraneous solutions are solutions that are excluded from the domain of the original equation.
Exclusions
The given equation is undefined for x an integer multiple of π, that is, for x=0 or x=π. Hence these values are excluded from the domain of the equation. If these show up as a solution, they will be extraneous solutions.
The domain of the equation excludes x = 0 and x = π.
Solution
Multiplying the equation by sin(x), we have ...
cos(x)(cos(x)/sin(x))·sin(x) = (3 -3sin(x))·sin(x) . . . . . multiply by sin(x)
cos(x)² = 3sin(x) -3sin(x)² . . . . . . . . simplify
1 -sin(x)² -3sin(x) +3sin(x)² = 0 . . . . . . . subtract the right-side expression
2sin(x)² -3sin(x) +1 = 0 . . . . . . . . . write in standard form
(2sin(x) -1)(sin(x) -1) = 0 . . . . . . . factor
sin(x) = 1/2 or 1 . . . . . . values of sin(x) that make the product 0
x = π/6, π/2, 5π/6 . . . . . values of x that are solutions
Note that none of these values of x are excluded from the domain, so there are no extraneous solutions.
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Additional comment
Rewriting the equation to the form f(x)=0, the solutions are the x-intercepts of the graph of f(x). These are shown in the attachment. That graph shows the equation is undefined at x=0 and x=π, but those are not potential solutions. None of the solutions we found are extraneous.
Note that this method of solving a equation, finding the zeros of f(x)=0, is usually a good way to avoid extraneous solutions. A graphing calculator cannot show solutions that are not in the domain of f(x).
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