Mathematics-3

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



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