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Trig identities
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- 2. What is tobe learned? • Some new trig formulae • Correct methodology and structure for proofs
- 3. Squaring Sinx Sinx XSinx Written as sin2 x =(Sinx)2
- 4. Some exciting calculations sin2 (30)+ cos2 (30) (sin30)2 + (cos30)2 Repeat for 75.30 Rule sin2 x + cos2 x = 1 = 1
- 5. Two “new” rules sin2 x+ cos2 x = 1 so sin2 x = 1 and cos2 x = 1 – cos2 x – sin2 x
- 6. = 1 –cos2 x = 1 – sin2 x sin2 x + cos2 x = 1 Memory Test sin2 x cos2 x
- 7. = 1 –cos2 x = 1 – sin2 x sin2 x + cos2 x = 1 Memory Test 2 sin2 x cos2 x
- 8. sin2 x + cos2 x+ 4 Simplifying 1 + 4 = 5 (sin2 x + cos2 x)5 – 15 – = 4
- 9. Simplifying (sin2 x + cos2 x) X1 5 5 5sin2 x + 5cos2 x = 5
- 10. Simplifying 1 – sin2 x+ 5cos2 x cos2 x + 5cos2 x = 6cos2 x
- 11. Trig Identities sin2 x +cos2 x = 1 so sin2 x = 1 – cos2 x and cos2 x = 1 – sin2 x Blank Space
- 12. Simplifying (sin2 x + cos2 x) X1 7 7 7sin2 x + 7cos2 x = 7 1 – cos2 x + 3sin2 x sin2 x + 3sin2 x = 4sin2 x
- 13. Almost as exciting sin600 ÷cos600 tan600 sin200 ÷ cos200 tan200 Rule tanx = sinxcosx 1.73 1.73 0.36 0.36
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- 15. = 1 –cos2 x = 1 – sin2 x sin2 x + cos2 x = 1 Memory Test 2 sin2 x cos2 x Sinx Cosx Tanx =
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- 17. Trig Identities sin2 x +cos2 x = 1 so sin2 x = 1 – cos2 x and cos2 x = 1 – sin2 x Tanx = Sinx Cosx
- 18. Proof Type Questions Prove2sin2 x+ 2cos2 x = 2 LHS 2sin2 x+ 2cos2 x = 2(sin2 x + cos2 x) = 2 X 1 = 2 sin2 x + cos2 x = 1 sin2 x = 1- cos2 x cos2 x = 1 - sin2 x( ) sinx = tanx =RHS QED cosx quod erat demonstrandum
- 19. Prove 1 –cos2 x = tan2 x LHS 1 – cos2 x sin2 x sin2 x + cos2 x = 1 sin2 x = 1- cos2 x cos2 x = 1 - sin2 x( ) =RHS QED coscos22 xx coscos22 xx cos2 x = tan= tan22 xx sinx = tanxcosx Key Question