# 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 *P*_{a} 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
*P*_{a} 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 3*t* and *y* = cos 2*t*, 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?

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