4. Applications: Derivatives of Trigonometric Functions

by M. Bourne

We can now use derivatives of trigonometric and inverse trigonometric functions to solve various types of problems.

Example 1

Find the equation of the normal to the curve of `y=tan^-1x/2` at `x=3`.

Example 2

The apparent power Pa of an electric circuit whose power is P and whose impedance phase angle is θ, is given by

`P_a = P\ sec\ θ`.

Given that P is constant at 12 W, find the time rate of change of Pa if θ is changing at the rate of 0.050 rad/min, when θ = 40.0°.

Example 3

A machine is programmed to move an etching tool such that the position of the tool is given by x = 2 cos 3t and y = cos 2t, where the dimensions are in cm and time is in s. Find the velocity of the tool for t = 4.1 s.

Example 4

The television screen at a sports arena is vertical and 2.4 m high. The lower edge is 8.5 m above an observer's eye level. If the best view of the screen is obtained when the angle subtended by the screen at eye level is a maximum, how far from directly below the screen must the observer be?

Example 5

A winch on a loading dock is used to drag a container along the ground. The winch winds the cable in at 2ms-1 and is 5 m above the ground. At what rate is the angle θ between the cable and the ground changing when 10 m of cable is out?