On-line Math 21

3.1  Trigonometric functions

3.1.1  Derivatives of sin(x) and cos(x) .

Example 4 Find the tangent line to the curve y = sin(x) at (p/6,1/2) , and use that to find, approximately, sin(p/6+0.1) .

Solution

You may want to glance back at the section on linear approximation first.

f(x) = sin(x) , so f¢(x) = cos(x) , and f¢(p/6) = Ö3/2 = 0.86602540378 , so the tangent line is y = f(a)+f¢(a)(x-a) = 1/2+Ö3/2(x-p/6) . The approximate value of sin(p+0.1) is then:

f(x) » f(a)+f¢(a)(x-a) sin(x) » 1 2 +Ö3/2(x-p/6) sin(p/6+0.1) » 1 2 +Ö3/2(0.1) sin(p/6+0.1) » 1 2 +0.86602540378(0.1) = 0.5866025403784438

On the other hand, the machine finds its own (similarly determined) value of sin(p/6+0.1) as sin(p/6+0.1) » 0.5839603578 . Which is closer?

Copyright (c) 2000 by David L. Johnson.