Question 1138302: Please help, I have no idea how to do this.
Find the general solution of sec3x=cosec2x
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if you know what the trigonometric identify formulas are, you can solve this.
otherwise, you can solve it by graphing, if you know how to graph them.
i solved by grphing first, using the desmos.com calculator.
here's what the calculator shows me.
in the interval between 0 and 360 degrees, the calculator tells me that the soluton is:
x = 18, 162, 234, 306.
subtract 360 from all of those and the calculator tells me that, in the interval from -360 to 0, the solution is:
x = -54, -126, -198, -342.
these solutions repeat every 360 degrees, therefore, the general solution would be:
18 plus or minus k * 360
162 plus or minus k * 360
234 plus or minus k * 360
306 plus or minus k * 360
k is an integer that is greater than or equal to 0.
if you wish to solve it algebraically, then it's good to know the trigonometric identities involved.
a pretty exhaustive list of trigonometric ikdentities can be found at https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle,_triple-angle,_and_half-angle_formulae
here's a display of the section in the reference that addresses double and triple angle formulas.
that section is about halfway down in the page of the reference.
it helps to know that secant is equal to 1 / cosine and that cosecant is equal to 1 divided by sine.
your equation of secant(3x) = cosecant(2x) becomes:
1 / cosine(3x) = 1 / sine(2x)
cross multiply to get sine(2x) = cosine(3x)
from the identity formulas, you get:
sine(2x) = 2 * sine(x) * cosine(x)
cosine(3x) = 4 * cosine^3(x) - 3 * cosine(x)
sine(2x) = cosine(3x) becomes:
2 * sine(x) * cosine(x) = 4 * cosine^3(x) - 3 * cosine(x)
divide both sides of this equation by cosine(x) to get:
2 * sine(x) = 4 * cosine^2(x) - 3
since cosine^2(x) is equal to 1 - sine^2(x), the equation becomes:
2 * sine(x) = 4 * (1 - sine^2(x)) - 3
simplify to get:
2 * sine(x) = 4 - 4 * sine^2(x) - 3
combine like terms to get:
2 * sine(x) = -4 * sine^2(x) + 1
add 4 * sine^2(x) to both sides of the equation and subtract 1 from both sides of the equation to and order the terms in descending order of degree to get:
4 * sine^2(x) + 2 * sine(x) - 1 = 0
if you let y = sine(x), this equation becomes:
4 * y^2 + 2 * y - 1 = 0
solve this quadratic equation to get:
y = -0.80901699437495 or y = 0.30901699437495
since y = sine(x), you get:
sine(x) = -0.80901699437495 or sine(x) = 0.30901699437495
solve for x to get:
x = -54 degrees or x = 18 degrees.
if you add 360 to -54 degrees, it will become the positive angle of 306 degrees.
the equivalwent angle in the first quadrant would be 360 - 306 = 54 degrees.
your possible solutions between 0 and 360 degrees would be:
18 and 54 in the first quadrant.
180 - 18 = 162 and 180 - 54 = 126 in the second quadrant.
180 + 18 = 198 and 180 + 54 = 234 in the third quadrant.
360 - 18 = 342 and 360 - 54 = 306 in the fourth quadrant.
you would want to evaluate 1 / cosine(3x) and 1 / sine(2x) in each of these angles to see which ones hold true.
18 is good. *****
54 is no good.
162 is good. *****
126 is no good.
198 is no good.
234 is good. *****
342 is no good.
306 is good. *****
the good angles are 18, 162, 234, 306.
these angles repeat every 360 degrees, therefore the solution is:
18 plus or minus 360 * k
162 plus or minus 360 * k
234 plus or minus 360 * k
306 plus or minus 360 * k
this agrees with the graphical solution.
if you need the answer in radians, just multiply the degrees by pi / 180 and you get the equivalent answer in radians.
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