Linearization problem

Question

Use a linear approximation to approximate sin(26°).

Let f(x)=sin(x). Find the equation of the line tangent of f(x) at a "nice" point near 26°. Then use the tangent line to approximate sin(26°). (the problem gave the hint to convert the degrees to radians)

Answer 1

A good "nice" point near 26° is 30°. Since sin 30° = 0.5 and cos 30° = (√3)/2 = 0.8660, the equation of the tangent line is y - 0.5 = 0.8660 × (π/180) × (x - 30). There is no need to convert the degrees to radians as long as you remember that factor π/180. To approximate sin 26° using this line, we simply substitute 26 for x in this equation and find the resulting value of y. I get sin 26° ≈ 0.5 - (0.8660 × π/45) = 0.4395. My calculator also gives the "exact" value of sin 26° as 0.4384.