Question

What is the error due to using linear interpolation to estimate the value of sinxsin⁡x at x = \pi/3? your answer should have at least three significant figures, accurate to within 0.1%. (e.g., 1.23 and 3.33e-8 both have three significant figures.)?

Answer 1

Answer:

• using y = x, the error is about 0.1812
• using y = (x -π/4 +1)/√2, the error is about 0.02620

Step-by-step explanation:

The actual value of sin(π/3) is (√3)/2 ≈ 0.86602540.

If the sine function is approximated by y=x (no error at x = 0), then the error at x=π/3 is ...

... x -sin(x) @ x=π/3

... π/3 -(√3)/2 ≈ 0.18117215 ≈ 0.1812

You know right away this is a bad approximation, because the approximate value is π/3 ≈ 1.04719755, a value greater than 1. The range of the sine function is [-1, 1] so there will be no values greater than 1.

___

If the sine function is approximated by y=(x+1-π/4)/√2 (no error at x=π/4), then the error at x=π/3 is ...

... (x+1-π/4)/√2 -sin(x) @ x=π/3

... (π/12 +1)/√2 -(√3)/2 ≈ 0.026201500 ≈ 0.02620