Finding roots for trigonometric function

Recall the non-polynomial function which was given and initial value:

Initial value

Using WZGrapher was made the graph for non-polynomial function, which shows bellow.

Graph 4

Graph 2 visually shows that roots should be 3, the first root is approximately between -2.579 and -2.553, the second root as approximately between -2.216 and -2.215, third is approximately between -2.049 and -2.028.

Using iterative method:

The first iterative way:

Assume that function of above is true, will get that:

Consider whether the first initial way works for non-polynomial function by substitution the initial value. Will get:

That’s mean, roots cannot be found by using first way. Hence that will be chosen another way.

The second iterative way:

Assume that function of above is true, will get that:

Using formula above and initial value was made Table 3.

Table 3

Step	Iterative
	x	g(x)
1	-2.000000	-2.135873
2	-2.135873	-2.027549
3	-2.027549	-2.070231
4	-2.070231	-2.009422
5	-2.009422	-2.111380
6	-2.111380	-2.004664
7	-2.004664	-2.123500
8	-2.123500	-2.013644
9	-2.013644	-2.101067
10	-2.101067	-2.000721
11	-2.000721	-2.133930
12	-2.133930	-2.025055
13	-2.025055	-2.075396
14	-2.075396	-2.005833
15	-2.005833	-2.120474
16	-2.120474	-2.010967
17	-2.010967	-2.107557
18	-2.107557	-2.002803
19	-2.002803	-2.128380
20	-2.128380	-2.018565
21	-2.018565	-2.089594
22	-2.089594	-2.000373
23	-2.000373	-2.134865
24	-2.134865	-2.026241
25	-2.026241	-2.072919

Table 3 was made by using Microsoft Excel.

Table above shows that first 25 roots are random numbers. Maybe it is possible to find root by using same way on calculator, but it will take a lot of time. Therefore cannot be sure that root will be correct. Hence, should be chosen another initial value.

Using Newton-Raphson method

First of all, should be found the first derivative of the trigonometric function:

So Newton-Raphson method to non-polynomial function:

Initial value is .

Using information above, was made Table 4.

Table 4

Step	Newton-Raphson method
	x	f(x)	f'(x)
1	2.0000000	0.6793666	-8.5177704
2	2.0797587	-0.3814667	2.6357595
3	2.2244861	0.0987751	21.5792596
4	2.2199088	0.0464554	21.2765272
5	2.2177254	0.0220013	21.1223592
6	2.2166838	0.0104554	21.0466152
7	2.2161870	0.0049768	21.0099936
8	2.2159502	0.0023709	20.9924184
9	2.2158372	0.0011299	20.9840133
10	2.2157834	0.0005386	20.9800004
11	2.2157577	0.0002567	20.9780859
12	2.2157455	0.0001224	20.9771729
13	2.2157396	0.0000583	20.9767376
14	2.2157369	0.0000278	20.9765300
15	2.2157355	0.0000133	20.9764311
16	2.2157349	0.0000063	20.9763839
17	2.2157346	0.0000030	20.9763614
18	2.2157344	0.0000014	20.9763507
19	2.2157344	0.0000007	20.9763456
20	2.2157343	0.0000003	20.9763432
21	2.2157343	0.0000002	20.9763420
22	2.2157343	0.0000001	20.9763415
23	2.2157343	0.0000000	20.9763412

Table 4 was made by using Microsoft Excel.

Table above shows the root, which value is . Therefore, in Graph 2 was visually shown that roots should be negative, not positive. According to this, root is inversed to the real root.

In summary, Iterative method, with given initial value does not work, while Newton-Raphson method found the rearward to real root.

Therefore, should be found new value of initial value to find roots for non-polynomial function.

New initial value to find the roots (Iterative method).

Now will be find new initial value, which will help to find the roots.

Graph 2, shows between what values should be chosen the roots. To check the range of values, will be used the change of sign method. First root should be chosen between -2.215 and -2.216. That’s mean and . Calculating theory above by calculating values of :

and

That’s mean the initial value was chosen correctly.

Hence that, guessing the value of root is between -2.215 and -2.216.

Using Microsoft Excel making new Table with new initial value to Initial and Newton-Raphson method.

Table 5

Step	Iterevative
	x	g(x)
1	-2.215000	-2.213392
2	-2.213392	-2.208292
3	-2.208292	-2.192384
4	-2.192384	-2.145725
5	-2.145725	-2.041929
6	-2.041929	-2.043771
7	-2.043771	-2.040804
8	-2.040804	-2.045631
9	-2.045631	-2.037908
10	-2.037908	-2.050587
11	-2.050587	-2.030689
12	-2.030689	-2.063965
13	-2.063965	-2.014908
14	-2.014908	-2.098064
15	-2.098064	-2.000218
16	-2.000218	-2.135284
17	-2.135284	-2.026781
18	-2.026781	-2.071804
19	-2.071804	-2.008239
20	-2.008239	-2.114344
21	-2.114344	-2.006429
22	-2.006429	-2.118943
23	-2.118943	-2.009722

Table above, shows how to find the root for non-polynomial function, using the new initial value. Hence, the first 23 number are showing as a “periodic”, root of function cannot be found. That means iterative method does not work for this no-polynomial function.

Can be adopted that the Iterative method does not work for non-polynomial function, while the Newton-Raphson works.