Math for AVT Dr I' Drumm

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Math for AVTDr I. Drumm

• Week 7
• Aims
• Introduce basic trigonometry and its role in
  acoustics
• Learning Outcomes
• The right angled triangle, basis trig functions
  (sin, cos, tan), degrees and radians,
  periodicity, sound as a sinusoid.

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Consider right angled triangle

• sin(?) opposite / hypotenuse
• cos(?) adjacent / hypotenuse
• tan(?) opposite / adjacent
• Pythagoras theorem

3
Evaluate trigonometric ratios

• Consider Isosceles triangle (two equal sides)

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Note about calculators
Make sure you are using appropriate mode, i.e.
degrees or radians Use inv function key to find
angle from sin, cos, tan
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Inverse functions

• Note inverse functions written as
• This is not the same as

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Solving right angled triangles

• Finding all angles and the lengths of all sides
  given some information
• For example
• if we know h ? then yhsin? and xhcos?
• if we know x ? then yxtan?
• if we know x and y then ?tan-1(y/x)
• We can also apply Pythagoras's theorem when
  needed

Exercises
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Trigonometric Functions

• Radians
• Represented by Anticlockwise around circle from 3
  o'clock position
• Clockwise is negative sense

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• cos?projection OQ of OP along x axis
• sin?projection OR of OP along y axis
• First quadrant
• ? 0 to 900 sin 0 to 1 cos 1 to 0
• Second quadrant
• ? 900 to1800 sin 1 to 0 cos 0 to -1
• Third quadrant
• ? 1800 to 2700 sin 0 to -1 cos -1 to 0
• Forth quadrant
• ? 2700 to 3600 sin -1 to 0 cos 0 to 1

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• Whats sin(120)
• sin(120) sin(180-60) sin(60)
• cos(120) -cos(180-60) -cos(60)
• Whats cos(300)
• sin(300) -sin(60)
• cos(300) cos(60)
• Note periodicity

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Graphs of trig functions
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Sound Generation
Oscillation is competition between stiffness an
inertia. In simplest case resulting motion is
sinusoidal (pure tone)
Sound propagation is the movement of sound energy
between elements of elastic medium (e.g. air)
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Sound as a sinusoid

• A is max amplitude
• is phase angle in radians

T is period f is frequency ? is angular
frequency