Let P be an approximation of pai accurate to n decimals. Show that P + sin⁡ P gives an approximation correct to 3n decimals. (Hint: Let P=p + x.)

Question

Let P be an approximation of pai accurate to n decimals. Show that P + sin⁡ P gives an approximation correct to 3n decimals.  (Hint: Let P=p + x.)

Mitchel Aguirre · Accepted Answer

Substitute the value of P ∈ |f(x)|. P=π + x |f(x)|=|P + sin⁡(P) − π| =|π + x + sin⁡(π + x)| =|π + x + sin⁡(π)cos⁡ x + cos⁡ π sin⁡ x − π| =|x + 0 + (−1)sin⁡x| =|x − sin⁡ x| P is accurate to n decimals so, |x|≤10−n −10−n≤ x ≤ 10−n x ∈[−10−n, 10−n] Use the Maclaurin Series of sin x expansion. sin⁡(x)=x − x33! + x55!⋯ |f(x)|=|x − sin⁡ x| =∣ x − (x − x33! + x55!⋯)∣ =∣x33! − x55!⋯∣ &amp;lt; x33! ∣x33! − x55!⋯∣ &amp;lt; x33!  &amp;lt; 10−n33!  &amp;lt; 10−3n

Let P be an approximation of pai accurate to n decimals. Show that P + sin P gives an approximation correct to 3n decimals. (Hint: Let P = p + x.)

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