Q.64: Given that : sin 45° = 0.7071, sin50° = 0.7660, sin55° = 0.8192, sin 60° = 0.8660, find the value of sin 52°.
Answer:
Given that sin(45°) = 0.7071,
sin(50°) = 0.7660,
sin(55°) = 0.8192,
sin(60°) = 0.8660.
Then find sin(52°).
We will use Newton's forward difference formula with x = 52.
Let x be the angle and also y = f(x) = sin x
Now xo = 45 and h = 5
So
x = xo + ph = 52
Now p = x – xo / h
= 52 – 45 / 5
= 1.4
Now we get the table as follows
So
x y Δy Δ^2y Δ^3y
45 0.7071
0.0509
50 0.7660 - 0.0057
0.0532 - 0.0007
55 0.8192 - 0.0064
0.0468
60 0.8660
Now by Newton – Gregory formula we have
Y1.4 = 0.7071 + (1.4) (0.0589) + 1.4 x 0.4 / 2 (- 0.0057) + 1.4 x 0.4 x – 0.6 / 6 (- 0.0007)
= 0.7071 + 0.08246 – 0.00160 + 0.0004
= 0.78836
= 0.7880 approximately
Therefore sin 52 = 0.7880