Angles measured in radians allow direct differentiation of trigonometric…

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Angles measured in radians allow direct differentiation of trigonometric relationships that connect angular quantities to linear measures. Answer the following questions about why radian measure is essential and how time–rate relationships follow from it. (a) Explain, without performing any calculation, why measuring an angle in degrees would complicate the differentiation of an expression such as

s=r\theta(t)

, where

r

is a fixed length and

\theta

is a time–dependent angle. (b) Starting from

s=r\theta(t)

with

\theta

measured in radians, use clear verbal reasoning to describe the exact role of the chain rule when relating

\dfrac{ds}{dt}

\dfrac{d\theta}{dt}

. (c) A cyclist rides along a circular track of radius 50 m so that the length of arc travelled after

t

seconds is

s(t)

. Describe how the signs of

\dfrac{ds}{dt}

and

\dfrac{d\theta}{dt}

reveal whether the cyclist is moving clockwise or counter-clockwise.

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