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Algebra for Athletes.com

How It’s Used

Land Surveyors are possibly the most frequent users of trigonometry.  They use it to find unknown distances and angles while measuring property and structures.

Trigonometry is often used to calculate distances.  But the sine wave is also found in many areas.  There are several examples of sinusoidal relationships in the physical world.  For example, when an object such as a satellite orbits a rotating planet, the projection of the satellite's path on the planet's surface is a sine wave. 

Sinusoidal motion is also common in vibrations.  If the spring in Fig. 12.25 is pulled down, it will oscillate up and down.  As shown in the figure, its height will have a sinusoidal relationship with time until it is brought to rest.  (When air friction slows a vibrating object, the vibration is said to be damped.  The affect of damping causes a logarithmic reduction in amplitude.) 

Virtually any flexible object which that is moved out of its normal position can function like a spring.  Figure 12.26 shows some objects that will vibrate if moved.  As it is pulled in one direction, the object will attempt to return to its original position.  As it springs back to its original position, it will gain some momentum and continue moving past its original position.  It will reach a point where forces will again try to return it to its original position.  It may take hundreds of cycles before the motion completely stops.  This type of motion causes buildings to oscillate after an earthquake.

Oscillatory motion is also behind the creation of music from virtually any type of instrument.  The strings on a violin, guitar, or piano will vibrate in the way described above.  Drums, woodwind, and brass instruments all generate sound waves from materials vibrating back and forth.

Another type of vibration regularly occurs with moving machines and vehicles.  Any type machinery that rotates or oscillates may vibrate in an undesirable way due to imperfections in its moving parts. For example, if the wheel on a car is out of alignment, it will create a vibration.  The frequency of the vibration will equal the frequency of the wheel rotation.  Similarly, if the blades on a ceiling fan are unbalanced, it will wobble when in use.  A typical form of an equation describing the displacement of a vibrating object has the form:

where       x = the displacement of the object at timet         (12 - 15)

               A = the maximum displacement

                   = the frequency of the vibration, in radians per second

                        t = time, in seconds 

Another field in which sinusoidal oscillation is common is electronics. In an AC (alternating current) circuit, the components are arranged in such a way that as the current increases, physical forces are created which cause the current to reverse direction.  After reversing direction, the current increases causing it to again reverse direction.  The process can happen thousands or millions of times per second.  A common form of an equation to express the electrical current, i, of an AC circuit is:

where  i = the current at time t, in amperes                           (12 - 16)

            A = the maximum current or amplitude, in amperes

       = the frequency of the current, in radians per second

            t = time, in seconds

As you might imagine, a great deal of trigonometry is required in the conversion between vibratory and electrical oscillations.  A microphone is an instrument that converts vibratory oscillations (from an instrument or a person’s voice) in to electrical oscillations.  A speaker converts electrical oscillations into vibratory oscillations which produce sound.