Question

Given Sin 20°=0.3420, Sin 25° =0.4426 Sin 30°=0.5000, Sin 35 -0.5736, Store Sin 40°=0.6428. Find & Sin 24 using 2 ° ising appropriate interpolation formulae.​

Answer 1

Sin 24 = 0.4221 when by using interpolation formula for expressions Sin 20°= 0.3420, Sin 25° = 0.4426, Sin 30°= 0.5000, Sin 35° = 0.5736.

Given that,

Sin 20°= 0.3420,

Sin 25° = 0.4426

Sin 30°= 0.5000,

Sin 35° = 0.5736,

Sin 40°= 0.6428.

We have to find Sin 24°

We know that,

Let x be the angle and y =f(x) =sin x

Now x₀ = 20 and h = 5

So,

x = x₀ + ph = 24

Now, p = $\frac{x-x_0}{h}$

          = $\frac{24-20}{5}$

          = 0.8

Now we get the table as follows

So

x          y                  Δy                     Δ²y                 Δ³y             Δ⁴y

20      0.3420

                               0.1006

25      0.4426                                 - 0.0432

                               0.0574                                     0.0594

30      0.5000                                  0.0162                            -0.08

                              0.0736                                     -0.0206

35       0.5736                                 -0.0044

                              0.0692

40        0.6428

Now by Newton – Gregory interpolation formula we have

Y(0.8) = 0.3420 + (0.8) (0.1006) + $\frac{0.8\times -0.2}{2}$ (- 0.0432) + $\frac{0.8\times-0.2\times -0.8}{6}$(0.0594) + $\frac{0.8\times-0.2\times -0.8\times -1.8}{24}$(-0.08)

       = 0.3420 + 0.08048 – 0.0003456 + 0.00012672 + 0.0000768

       = 0.4221

Therefore, The value of angle sin 24 = 0.4221

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