Question 677715: Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)
csc2 θ = 2 cot θ + 4
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! I'm assuming that the equation is
csc^2(θ) = 2cot(θ) + 4
and not
csc(2θ) = 2 cot(θ) + 4
If I am correct, then please use "^" to indicate exponents. If I am wrong then you will need to re-post your question.
Solving these equations usually starts with using algebra and/or Trig. identities to transform the equation so that you have one or more equations of the form:
trigfunction(expression) = number
It is not always easy to see how to make the desired transformation. When it is not obvious, then start by using Trig identities to reduce the number of different arguments or functions in the equation. In this equation the arguments are all θ's. So we do not need to reduce the number of arguments. But we do have two different functions, csc and cot. And we have a Trig identity that connect these two functions:
csc^2(θ) = cot^2(θ) + 1
Substituting the right side of this equation for the left side of your equation we get:
cot^2(θ) + 1 = 2 cot(θ) + 4
Now that we have just one function, cot, and one argument, θ, we are ready to find the desired form. Subtracting the entire right side from both sides we get:
cot^2(θ) - 2cot(θ) - 3 = 0
The left side will factor:
(cot(θ)-3)(cot(θ) + 1) = 0
Using the Zero Product Property we get:
cot(θ)-3 = 0 or cot(θ) + 1 = 0
Adding 3 to each side of the first equation and subtracting 1 from each side of the other we get:
cot(θ) = 3 or cot(θ) = -1
Both of these equations are now in the desired form.
The next step is to write the general solution for each equation. The general solution expresses all the solutions to the equations. We'll start with:
cot(θ) = 3
We should recognize that 3 is a not a special angle value for cot. So we will need our calculators. And since your calculator probably does not have cot buttons, we must convert this to tan. tan is the reciprocal of cot so if cot = 3 then...
tan(θ) = 1/3
Now we can use the inverse tan, tan^-1(1/3), to find the reference angle. We should get a reference angle of 18.43494882 degrees. With this reference angle and since cot (and tan) are positive in the 1st and 3rd quadrants we should get a general solution of
θ = 18.43494882 + 360k
and
θ = 180 + 18.43494882 + 360k
The second equation simplifies to:
θ = 198.43494882 + 360k
Now for
cot(θ) = -1
We should recognize that -1 is a a special angle value for cot. So we will not need our calculators. The reference angle for this is 45 degrees. With this reference angle and since cot is negative in the 2nd and 4th quadrants, we should get a general solution of
θ = 180 - 45 + 360k
and
θ = -45 + 360k (or 360 - 45 + 360k)
which simplify to:
θ = 135 + 360k
and
θ = -45 + 360k (or 315 + 360k)
Many, but not all, of these problems ask you to find a specific solution. For example: "Find the least positive solution to ..." or "Find all solutions to ... that are between zero and 360". When specific solutions are requested you use the general solution equation(s) and various integer values for "k" until you have found the requested solution(s).
In this case the problem does not ask for a specific solution. The general solution is the solution when no specific solution is requested. Putting the general solution into the requested form:
θ = 18.435 + 360k, 198.435 + 360k, 135 + 360k, -45 + 360k
P.S. The period for tan and cot is 180. So the solutions can be condensed to:
θ = 18.435 + 180k, 135 + 180k
(Note how the 360's have been replaced with 180's.)
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