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Chapter two

Chapter two. Trigonometry. NEGATIVE ANGLE IDENTITIES. sin(-θ) = - sinθ cos(- θ ) = cos θ tan(- θ ) = - tan θ csc(-θ ) = - cscθ sec(- θ ) = sec θ cot(- θ ) = - cot θ. Periodic Functions.

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Chapter two

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  1. Chapter two Trigonometry

  2. NEGATIVE ANGLE IDENTITIES • sin(-θ) = - sinθ • cos(-θ) = cosθ • tan(-θ ) = - tanθ • csc(-θ ) = - cscθ • sec(-θ) = secθ • cot(-θ ) = - cotθ

  3. Periodic Functions • Periodic Function- a function whose graph has a repeating pattern that continues indefinitely. The shortest repeating portion is called a cycle. • Period- The horizontal length of each cycle is called the period.

  4. Examples of periodic functions

  5. Symmetric about the y axis FUNCTIONS Symmetric about the origin

  6. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Even functions have y-axis Symmetry 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7 So for an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.

  7. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Odd functions have origin Symmetry 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7 So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph.

  8. A function is even if f( -x) = f(x) for every number x in the domain. So if you plug a –x into the function and you get the original function back again it is even. Is this function even? YES Is this function even? NO

  9. A function is odd if f( -x) = - f(x) for every number x in the domain. So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd. Is this function odd? NO Is this function odd? YES

  10. If a function is not even or odd we just say neither (meaning neither even nor odd) Determine if the following functions are even, odd or neither. Not the original and all terms didn’t change signs, so NEITHER. Got f(x) back so EVEN.

  11. Asymptotes • Vertical Asymptotes: Will occur at the x values that make the denominator 0.

  12. Horizontal Asymptotes • If the degree of the numerator is less than the degree of the denominator. Asymptote Y=0 • If the degree of the numerator is equal tothe degree of the denominator. Asymptote is the horizontal line y = a/b • When the degree of the numerator is greater than the degree of the denominator there is no horizontal asymptote!!

  13. Finding Horizontal Asymptotes If

  14. Graph of Example 4 The horizontal line y = 0 is the horizontal asymptote.

  15. Finding Horizontal Asymptotes Example 5 If

  16. Graph of Example 5 The horizontal dotted line at y = 6/5 is the horizontal asymptote.

  17. Finding Horizontal Asymptotes Example 6 If There are no horizontal asymptotes because the degree of the numerator is greater than the degree of the denominator.

  18. Graph of Example 6

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