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Sagot :
The solution of the assumed equation is
θ = 135 + 360k
and
θ = -45 + 360k (or 315 + 360k)
Assuming the equation is
csc^2(θ) = 2cot(θ) + 4
and not
Assuming the equation to be:
csc^2(θ) = cot^2(θ) + 1
Solving these equations usually begins with algebra and/or trigonometry. ID for transforming equations to have one or more equations of the form:
trigfunction(expression) = number
It is not always easy to understand how to perform the desired transformation. If it's not obvious, first use the trigonometric identity to reduce the number of different arguments or functions in the equation. In this expression, all arguments are θ. Therefore, there is no need to reduce the number of arguments. But he has two different functions, csc and cot.
csc^2(θ) = cot^2(θ) + 1
Substituting the right side of this equation into the left side of the equation, we get: :
cot^2 (θ) + 1 = 2 cot(θ) + 4
Now that we have only one function cot and one argument θ, we are ready to find the form we need. Subtracting the entire right-hand side from both sides gives:
cot^2(θ) - 2cot(θ) - 3 = 0
The left-hand side factorizes:
(cot(θ)-3)(cot (θ) ) + 1 ) = 0
Using the properties of the zero product,
cot(θ) = 3 or cot(θ) = -1
These two equations are now in the desired form.
The next step is to write the general solution for each equation. The general solution represents all solutions of the equation.
cot(θ) = 3
Note that 3 is not a specific angle value for cot. That's why you need a calculator. Your computer probably doesn't have the crib button, so you'll need to switch it to tan
Since tan is the reciprocal of cot, if cot = 3...
tan(θ) = 1/3
Inverse tan tan^-1(1/3) can be used to find the reference angle. You should get a reference angle of 18.43494882 degrees. Using this reference angle and cot (and tan) being positive in the 1st and 3rd quadrants, we get the general solutions
θ = 18.43494882 + 360k
and
θ = 180 + 18.43494882 + 360k
.
θ = 198.43494882 + 360k
where
cot(θ) = -1
-1 should be recognized as a special angle value for cot. So you don't need a calculator. This reference angle is 45 degrees. Using this reference angle, cot is negative in the 2nd and 4th quadrants, so
θ = 180 - 45 + 360k
and
θ = -45 + 360k (or 360 - 45 + 360k) the general solution for
must get. to:
θ = 135 + 360k
and
θ = -45 + 360k (or 315 + 360k)
For more information about trigonometric identities, visit
https://brainly.com/question/24496175
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