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Trigonometry

Trigonometry. Trigonometric Identities. Trigonometric Identities.  An identity is an equation which is true for all values of the variable.  There are many trig identities that are useful in changing the appearance of an expression.

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Trigonometry

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  1. Trigonometry Trigonometric Identities

  2. Trigonometric Identities  An identity is an equation which is true for all values of the variable.  There are many trig identities that are useful in changing the appearance of an expression.  The most important ones should be committed to memory.

  3. Trigonometric Identities Reciprocal Identities Quotient Identities

  4. cos2θ + sin2θ = 1 By Pythagoras’ Theorem x2 + y2 = r2 (x, y) r y Divide both sides by r θ x

  5. Trigonometric Identities  Pythagorean Identities  The fundamental Pythagorean identity  Divide by sin2 x  Divide by cos2 x

  6. Identities involving Cosine Rule  Using the usual notation for a triangle, prove that c(bcosA–acosB) = b2–a2

  7. Identities involving Cosine Rule  Using the usual notation for a triangle, prove that c(bcosA–acosB) = b2–a2

  8. Trigonometric Formulas Page 9 of tables

  9. Trigonometric Formulas

  10. Replace B with – B

  11. Replace B with A

  12. Replace A with – A

  13. Solving Trig Equations •  To solve trigonometric equations: •  If there is more than one trigonometric function, • use identities to simplify •  Let a variable represent the remaining function •  Solve the equation for this new variable •  Reinsert the trigonometric function •  Determine the argument which will produce the • desired value

  14. 1 2 1 2 cos2A = (1 + cos 2A). cos2A = (1 + cos 2A) (i) Using cos 2A = cos2A – sin2A, or otherwise, prove cos 2A = cos2A – (1 – cos2A) sin2A cos 2A = cos2A – 1 + cos2A 1 + cos 2A = 2cos2A – 1 2005 Paper 2 Q4 (b)

  15. 1 –1 180º 360º (ii) Hence, or otherwise, solve the equation 1 + cos2x = cosx, where 0º ≤ x ≤ 360º. 1 + cos2x 2cos2x = cosx From (i) 2cos2x –cosx = 0 cos x(2cosx – 1) = 0 cos x = 0 2005 Paper 2 Q4 (b)

  16. Expand Collect like terms Rearrange Factorise

  17. 1 π 2π –1 Replace t with sinx

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