What is the linear approximation to sin

Paul Duran

Paul Duran

Answered question

2022-04-12

What is the linear approximation to sin ( 0.5 )?

Answer & Explanation

Charlie Powers

Charlie Powers

Beginner2022-04-13Added 13 answers

The formula for linear approximation is f ( x ) f ( a ) + f ( a ) ( x a ) .. Using f ( x ) = sin x this becomes sin x sin a + cos a ( x a ) ..
We can choose any center, any a value we like. You suggest 0 and π / 3 as options. I would suggest we also consider π / 6 ,, since that is a special angle that is close to 0.5.. Let's see:
sin ( 0.5 ) sin ( π / 3 ) + cos ( π / 3 ) ( 0.5 π / 3 ) 0.0924266 sin ( 0.5 ) sin 0 + cos 0 ( 0.5 0 ) = 0.5 sin ( 0.5 ) sin ( π / 6 ) + cos ( π / 6 ) ( 0.5 ( π / 6 ) ) = 1 2 + 3 2 ( π 6 1 2 ) 0.479563
And my calculator tells me sin ( 0.5 ) 0.479426. So our linearization at a = π / 3 does not approximate the value of sin ( 0.5 ) very well. The linearization at a = 0 matches to one decimal place. And the linearization at a = π / 6 matches to four decimal places. So that one is the closest. To get the best approximation, you should choose the center for your linearization closest to your input value.

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