OCR A LEVEL 2023 MATHEMATICS B PAPER 3 INSERT
A Level Mathematics B (MEI)
H640/03 Pure Mathematics and Comprehension
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Approximating the sine function
Small angles
For a small angle x radians, the approximation sin x . x is valid. The curve y = sin x and the
straight line y = x are shown in Fig. C1.1. Fig. C1.2 shows the curve y = x - sin x. Inspection of
the graphs suggests that x is a reasonable approximation for sin x for -0.5 G x G 0.5 and also that
y = x has the same gradient as y = sin x when x = 0. 5
y y
3
2
1
x
–2 –1 0
–1
x
1 2 3 4 5
Fig. C1.1
–2
Fig. C1.2
Calculating sin x
Trigonometric functions, including sin x, are widely used so it is useful to be able to calculate the value of the sine of any angle accurately and quickly. This is easily done nowadays using a calculator but this was not possible in the past. The linear function, y = x, is only a reasonable
approximation for y = sin x for values of x close to zero. Perhaps using a higher degree polynomial 10
would give a reasonable approximation for a wider range of values of x.
Fig. C2.1 shows the curve
y = sin x and the quadratic curve which goes through the points (0, 0),
ar, 1k and (r, 0). The equation of this curve is
y = 4x (r - x) - sin x.
r2
y = 4x (r - x) . Fig. C2.2 shows the curve
r2
y y
2 2
1 1
–2 –1 0
–1
x
1 2 3 4 5 –2
–1 0
–1
x
1 2 3 4 5
–2
Fig. C2.1
–2
Fig. C2.2
3
The quadratic function seems to be a reasonably good approximation for sin x in the interval 15 0 G x G r. However, calculating percentage errors for selected values of x shows that the percentage errors made by using the quadratic function as an approximation to sin x are quite high for values
of x close to zero or r.
The spreadsheet in Fig. C3 shows values of x in column A, with the corresponding values of sin x
and the quadratic function
4x (r - x)
r2
in columns B and C. Columns D and E show the percentage 20
errors in using x and the quadratic as approximations for sin x.
A B C D E
1 x sin(x) quadratic % error for x % error for quadratic
2 0 0 0
3 0.1 0.099833 0.123271 0.166861 23.476799
4 0.2 0.198669 0.238437 0.669791 20.016773
5 0.3 0.295520 0.345496 1.515901 16.911206
6 0.4 0.389418 0.444450 2.717298 14.131825
7
Fig. C3
A better approximation
The approximation
sin x . 16x (r - x)
5r2 - 4x (r - x)
was discovered by an Indian mathematician named
Bhaskara in the 7th century. It is not known how Bhaskara derived the formula but it can be seen that
the curve
y = 16x (r - x)
is symmetrical about x = r and goes through the points (0, 0), 25
5r2 - 4x (r - x)
2 16x (r - x)
ar, 1k and (r, 0). Fig. C4 shows the curves y = sin x and y =
5r2
- 4x (r - x)
. Radians were not in
use until the 18th century; Bhaskara gave the formula for an angle i degrees as
4i (180 - i) 40500 - i (180 - i)
y
2
1
–2 –1 0
–1
x
1 2 3 4 5
–2
Fig. C4
The percentage error in approximating sin x by
16x (r - x)
5r2 - 4x (r - x)
is less than 2% throughout the
interval 0 G x G r. The Bhaskara approximation for sin x can be used to derive the following 30
approximation for cos x; cos x . r2 - 4x2 .
r2 + x2 [Show Less]