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Understanding Trigonometric Ratios and Identities: A Comprehensive Guide

Explore the fundamentals of sine, cosine, tangent functions, and trigonometric identities. Learn how to express ratios in terms of sine and cosine, solve trigonometric equations, and prove key identities using various models. This comprehensive guide provides valuable insights into periodic and exponential functions.

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Understanding Trigonometric Ratios and Identities: A Comprehensive Guide

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  1. Topic 11Periodic and Exponential Functions I

  2. Recall • A S T C

  3. opp hyp + + sinθ= = = + cosθ= = = + tanθ= = = + adj hyp + + θ opp adj + +

  4. opp hyp + + sinθ= = = + cosθ= = = - tanθ= = = - adj hyp - + θ opp adj + -

  5. opp hyp - + sinθ= = = - cosθ= = = - tanθ= = = + adj hyp - + θ opp adj - -

  6. opp hyp - + sinθ= = = - cosθ= = = + tanθ= = = - adj hyp + + θ opp adj - +

  7. A S T C ll tations o entral SIN + ALL + TAN + COS +

  8. A S T C ll aints each lasses SIN + ALL + TAN + COS +

  9. A S T C ll heilas alk onstantly Yappity yappity… Blah blah blah !!!! SIN + ALL + TAN + COS +

  10. Sine of any angle Cosine of any angle Tangent of any angle

  11. Recall • A S T C • tan  = sin  / cos  • sin (90-) = cos  • cos (90-) = sin  • sin2 + cos2 =1 • sin, cos and tan of 30o , 60o and 45o

  12. 30 60 sin 30o = ½ cos 30o = tan 30o = sin 60o = cos 60o = ½ tan 60o =

  13. 45 45 sin 45o = cos 45o = tan 45o = 1

  14. So we can express every trig ratio in terms of sin and cos

  15. Model: Evaluate (a) cosec /4 (b) cot 5/3

  16. Model: If sec  = 5/4, find cot  if  is acute 5 4 3

  17. Model: Prove(a) cosec2x = sec x cosec x cot x(b) 1/(cosec x + 1) + 1/(cosec x -1) = 2tan x sec x

  18. Model: Prove(a) cosec2x = sec x cosec x cot x(b) 1/(cosec x + 1) + 1/(cosec x -1) = 2tan x sec x

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