Trigonometry Dr Ian Drumm

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TrigonometryDr Ian Drumm

• Week 8
• Aims
• Discuss trigonometric identities
• Learning Outcomes
• Addition Formulae, Half and Double angle
  Formulae, Sine and Cosine Rules

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Review Last Week

• Remember

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Consider unit circle and sine function
Note sin-1(y/1) has solutions ? and p-? (in
degrees ? and 180-?) Solutions repeat every
2p Similarly cos-1(x/1) has solutions ? and 2p-?
(in degrees ? and 360-?)
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Some key Trigonometric Identities

• Consider the tan in terms of sine and cosine
• Note also identities for cotangent, secant and
  cosecant

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• A trig identity is a relation that is always
  true, e.g. the following is true for all values
  of ?
• This identity is and application of Pythagoras
  theorem on the unit circle

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• Given

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Addition Formulae

• Useful to have identities for sin(AB) and
  cos(AB)
• Consider two right angle triangles, a third
  triangle can hence be constructed as shown

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Consider as on unit circle where angle CAB
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• Double angle formula
• Half angle formula

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Factor Formulae

• For adding two sinusoids
• Writing CAB and DA-B
• Similarly

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Non right angled triangles

• Whats the relationship between a, b and C?
• Consider area S of right angled triangle, given
  by base and height

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The Sine Rule

• For solving triangles
• Given
• Can show that

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Cosine Rule

• By considering triangle can also show

When an angle and the two sides forming it are
given use the cosine rule
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Solving triangles with sin rule

• Note all angles add up to 1800 (p radians)
• Note sin-1 can return ? and 180-? hence possible
  ambiguity

47.6100 or 180-47.61132.3900
Though 132 invalid because 13280gt180