Please Help me Solve this hard Maths equation for a
0-3+3cos(a/3)=-1/(sin(a/3)) * (9-a)
I have no idea how to do the working out and the answer for a is supposed to be a=5.41
Please Help me Solve this hard Maths equation for a
0-3+3cos(a/3)=-1/(sin(a/3)) * (9-a)
I have no idea how to do the working out and the answer for a is supposed to be a=5.41
Rounded to two places, 5.41 is one of the three solutions.
We can simplify the expressions above, by making a substitution.
Let k = a/3 (so a = 3*k)
This substitution leads to an equivalent equation:
1 - cos(k) = (3 - k)/sin(k)
Using an identity for sin(2*k) we also get:
sin(k) - 1/2*sin(2*k) = 3 - k
Graphing both sides of either of these equations shows three solutions for k (see below). Tripling each of these solutions yields the solutions for a.
When we have an equation comprised of a trigonometric function set equal to a polynomial function, it is often impossible to solve by hand.
I approximated the three solutions for k, by zooming in on the intersection points of a graph. One could also use a Computer Algebra System, to approximate the solutions.
There are methods for approximating by hand; have you heard of Newton's Method?
"English is the most ambiguous language in the world." ~ Yours Truly, 1969
I havent heard of Newtons method.
Is there anyway just to end up at 5.41 through algebra only.
Thanks
No. Your equation relates a trigonometric function (transcendental) to a linear function (polynomial). There's no closed-form approach, for finding solutions; we need to approximate the three solutions, using an iterative process of some sort.
Google the following, for more information:
difference between polynomial and transcendental equation
"English is the most ambiguous language in the world." ~ Yours Truly, 1969
ok thankyou
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