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8.4 Relationships Among Functions

8.4 Relationships Among Functions. Simplifying Trigonometric Identities. Reciprocal Functions. sin(- θ )=-sin θ csc(- θ )=-csc θ tan(- θ )=-tan θ. cos(- θ )=-cos θ sec(- θ )=-sec θ cot(- θ )=-cot θ. Relationships with Negatives. Pythagorean Relationships. sin 2 θ +cos 2 θ =1

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8.4 Relationships Among Functions

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  1. 8.4 Relationships Among Functions Simplifying Trigonometric Identities

  2. Reciprocal Functions

  3. sin(-θ)=-sinθ csc(-θ)=-cscθ tan(-θ)=-tanθ cos(-θ)=-cosθ sec(-θ)=-secθ cot(-θ)=-cotθ Relationships with Negatives

  4. Pythagorean Relationships • sin2θ+cos2θ=1 • 1+tan2θ=sec2θ • 1+cot2θ=csc2θ

  5. sinθ=cos(90°-θ) tanθ=cot(90°-θ) secθ=csc(90°-θ) cosθ=sin(90°-θ) cotθ=tan(90°-θ) cscθ=sec(90°-θ) Co function Relationships

  6. Guidelines for Simplifying Trigonometric Identities • Use the relationships • Try writing everything in terms of sine and cosine. • Try to factor, add fractions, FOIL or find the LCD. • Use algebra to simplify

  7. Example tan(x)cos(x) Writing in Terms of sin(x) and cos(x) =

  8. Examples cot(x)sin(x) 2) sec(x)cos(x) 3) 4)

  9. Examples 5) 6) 7)

  10. ASSIGNMENT • Page 320 #8 • Page 321 #1-12

  11. 8.4 Relationships Among Functions Verifying Trigonometric Identities

  12. Guidelines for Verifying Trigonometric Identities • Work with one side at a time, start with the more complicated side & note what functions you are trying to get to. • Try to factor, add fractions, FOIL or find the LCD. • When in doubt, try converting everything in terms of sin & cos. • DO NOT CROSS MULTIPLY OR ADD/SUBTRACT FROM BOTH SIDES: you can’t assume they are equal until you prove they are equal.

  13. LHS

  14. LHS

  15. LHS

  16. RHS

  17. Practice • secxcosx = 1 • cos2x – sin2x = 2cos2x – 1 • csc(x)sin(x) = 1 • (1 + cot²(x))(1 - sin²(x)) = cot²(x)

  18. ASSIGNMENT • Page 321 #13-24, 29-36

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