# 5. Applications of Trigonometric Graphs

by M. Bourne

Oscilloscope output - Filter modulation [Image source: Mikael Altemark]

## Simple Harmonic Motion

Any object moving with constant angular velocity or moving up and down with a regular motion can be described in terms of SIMPLE HARMONIC MOTION.

The displacement, *d*, of an object moving with SHM, is
given by:

d=Rsinωt

where R is the radius of the rotating object and `ω` is the angular velocity of the object.

For an animation of this concept, go back to: **sin animation**.

**NOTE:** We may need to use one the following,
depending on the situation:

### Need Graph Paper?

d=Rcosωt

d=Rsin (ωt+α)

d=Rcos (ωt+α)

### Example 1

A point on a cam is `8.30\ "cm"` from the centre of
rotation. Sketch 2 cycles of *d* as a function
of *t*, given that *d *= 0 cm when *t* = 0 s and *ω* = 3.20
rad/s.

### Example 2

The voltage of an alternating current circuit is given by

e=Ecos(ωt+α).

Sketch 2 cycles of the voltage as a function of time if

`E = 80\ "V"`, ` ω = 377\ "rad/s"` and `α = π/2`.

### Example 3

The signal received by a radio is given by

e= 0.014 cos(2πft),

where *e *is in volts and
*f* is in Hz.

Draw 2 cycles of *e *for
`f = 950\ "kHz"`.

## Angular Velocity

Another important result in this section is:

The **angular velocity** **ω** (in radians per second)
of a rotating object, is given by:

ω= 2πf

where *f* is the frequency of the motion, in cycles per
second.

### Exercises

1. A satellite is orbiting the earth so that its displacement
*D* north of the equator is given by

D=Asin(ωt+α).

Sketch 2 cycles of *D* as a function of *t* if

`A = 500\ "km"`, `ω = 3.60\ "rad/hr"` and ` α = 0`.

2. Using *e *= *E *cos(*ω**t* + *α*), sketch 2 cycles of the voltage as a
function of time if

E= 170 V,ω= 120πrad/s and `α = -π/3`.

### Search IntMath, blog and Forum

### Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Go to: Online algebra solver

### Trigonometry Lessons on DVD

Easy to understand trigonometry lessons on DVD. See samples before you commit.

More info: Trigonometry videos

### The IntMath Newsletter

Sign up for the free **IntMath Newsletter**. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!