How do you find the cos / sin of 2pi/5 by hand?

Question

no calculators, answer must be exact

Answer 1

Let x =72 degrees
5x = 360
2x = 360-3x
Taking cos both side
cos 2x = cos(360 -3x)
cos 2x = cos 3x
2cos^2 x -1= 4cos^3 x - 3 cos x
4cos^3 x - 2cos^2 x - 3 cos x +1=0
4cos^3 x - 4cos^2 x + 2 cos^2 x - 2 cos x - cos x +1=0
4cos^2x(cos x -1) +2cos x( cos x -1) -1( cos x - 1) =0
(cos x - 1){4cos^x +2 cos x - 1} =0
Either cos x =0 means x=90 degree
Solve ){4cos^x +2 cos x - 1} =0
cos x = [-2 +(plus or minus){sq rt (4+16)] /8
cos x = [-2 +(plus or minus){sq rt (20)] /8
cos x = [-2 +2(plus or minus){sq rt (5)] /8
cos x = [-1 +(plus or minus){sq rt (5)] /4
Neglecting -ve values of cos x, because x lies in first quadrant
There fore
cos 72^o= {sq rt(5) -1}/4
We can find the value of sin 72^o by using the formula sin^2 x + cos^2 x =1

Answer 2

I am offering you the most exciting solution... you difficultly will find it in Math books.

You must follow a procedure. Take a pencil, a ruler and a paper.

1) 2pi/5 radians is equivalent to 72 degrees
2) Draw a isosceles triangle with two angles of 72 degrees in the base... the vertex angle will be 36 degrees.

(You have got a marvel triangle... it is the golden triangle... it is called so because if you divide the side by the base(short side) you will find the golden number phi = 1.618... = (sqrt(5)+1)/2 ... and the marvel go far... if you draw the bissectrix of the 72 degree angle, you will find another golden triangle with the same angles.)

3) Now give values to the triangle : 10, 10, and x for the base
4) Because it is a golden triangle the ratio 10/x is 1.618...
or 10/x = (sqrt(5) + 1)/2 and from here you can get the x value;
you must find x = 5(sqrt(5) - 1)
5) Now you know the three sides. Draw the height from the vertex to the base. Will appear a rectangular triangle and the side of this rectangular triangle is x/2 = 5(sqrt(5)-1)/2
6) Now you can get cos(72degrees). It is the ratio between x/2 and 10. You will find (sqrt(5) -1)/4
7) The sin(72degrees) can be got by cos(x)^2 + sin(x)^2 = 1

Answer 3

If you had a perfect circle and could measure lengths exactly, then you could find these ratios by constructing right triangles inside the triangle. (cos and sin are just a ratio of sides)

Otherwise, there's no way to calculate it exactly by hand analytically. You can only approximate it to a certain number of digits.

One easy way to do it is to use the Taylor series expansion of cos (or sin) around x=0. You know that cos(0)=1, so it's easy to generate the expansion around x=0 (and similarly for sin(x) around x=0, since sin(0)=0).

cos(x) = 1 - x^2/(2!) + x^4/(4!) - x^6/(6!) + x^8/(8!) -+ . . .

sin(x) = x - x^3/(3!) + x^5/(5!) - x^7/(7!) + x^9/(9!) -+ . . .

Notice the ALTERNATING SIGN and notice the EVEN POWERS for cos and ODD POWERS for sin. Also note that the ! denotes a factorial, so:

8! = 8*7*6*5*4*3*2*1

and

4! = 4*3*2*1

In this case, the more terms you add, the closer you'll converge to the "actual" value. In these converging alternating series, if your next iteration is close enough to your previous iteration, you can stop. That is, once you've calculated the approximation for 5 terms and 6 terms, if the difference in that approximation is less than the error you can tolerate, then there is no need to add more terms.

Of course, to do this you would need a pretty good approximation for pi ( pi = 3.141592653589793... ) to plug into those polynomials.

Answer 4

Before the day of calculators, they used the Taylor Series approximations for trig functions.

Taylor series was very useful, because you could add up a small number of terms (usually 3 or 5 ) to get an answer accurate to the required number of sig-digs.

Answer 5

There are 2pi radians on the coordinate system.

If you sketch out an angle of 2pi/5 radians and drop a perpedicular to the x-axis, you will find cos and sin of the angle.

Answer 6

finding this value using Euclidean geometry is like trisecting an angle using ruler & divider (compass).... since 5 is the critical element here and since it is a prime, there would be no way to reach it by elementary geometrical operations...

but we can find using trigonometry....

now, 2 pi / 5 = 72 deg = x

let sin(x) = t

sin (5x) = sin (3x+2x) = sin3x.cos2x+cos3x.sin2x

but sin3x = sin2x.cosx+cos2x.sinx and so on

we can reduce the equation in terms of t (knowing that cosx=sqr(1-t^2)

sin5x=0 and hence we can find the value of 't', obviously!

please see: http://www.andrews.edu/~calkins/math/webtexts/numb18.htm where derivation of sin(18)=cos(72) is given

Answer 7

Consult a math book or your father. Guys usually know these things.