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sin
(
0
.
02
)
-
0
.
02
=
-
0
.
00
000
133
330
666
692
…
[1]
How many terms of Maclaurin series do I need
to calculate sin(x)-x at x=0.02 with the same result?
First term:
sin
(
0
)
-
0
=
0
.
0
[2]
Plus second term:
ans
+
(
cos
(
0
)
-
1
)
*
0
.
02
=
0
.
0
[3]
Plus third term:
ans
+
(
-
sin
(
0
)
)
/
2
*
0
.
02
*
*
2
=
0
.
0
[4]
Plus fourth term:
ans
-
cos
(
0
)
/
fact
(
3
)
*
0
.
02
*
*
3
=
-
0
.
00
000
133
333
333
333
…
[5]
Odd terms have sin(0) therefore they can be skipped.
Plus sixth term:
ans
+
cos
(
0
)
/
fact
(
5
)
*
0
.
02
*
*
5
=
-
0
.
00
000
133
330
666
667
…
[6]
Plus eighth term:
ans
-
cos
(
0
)
/
fact
(
7
)
*
0
.
02
*
*
7
=
-
0
.
00
000
133
330
666
692
…
[7]
Eight terms! So with eight terms, what is the error?
[1]
-
[7]
=
1
.
41
092
961
361
E-21
…
[8]
Of course much of this error is the 10th term:
cos
(
0
)
/
fact
(
9
)
*
0
.
02
*
*
9
=
1
.
41
093
474
427
E-21
…
[9]
CalculatorPi.com
1
2
3
+
π
sin
asin
(
)
4
5
6
−
e
cos
acos
exp
←
7
8
9
×
g
tan
atan
ln
,
•
0
E
∕
R
rad
deg
log(a,b)
ans
y
x
√
abs
round
N
rand
fact
mod(a,b)
CLR
Batch edit
sin(0.02) - 0.02 How many terms of Maclaurin series do I need to calculate sin(x)-x at x=0.02 with the same result? First term: sin(0) - 0 Plus second term: ans + (cos(0) - 1) * 0.02 Plus third term: ans + ( - sin(0)) / 2 * 0.02**2 Plus fourth term: ans - cos(0) / fact(3) * 0.02**3 Odd terms have sin(0) therefore they can be skipped. Plus sixth term: ans + cos(0) / fact(5) * 0.02**5 Plus eighth term: ans - cos(0) / fact(7) * 0.02**7 Eight terms! So with eight terms, what is the error? [1] - [7] Of course much of this error is the 10th term: cos(0) / fact(9) * 0.02**9
This month's formula: basic two vector operations.
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