Analytical Trigonometry

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  1. The distributive law for real numbers: r(s + t) =? However, if f is a function, and you replace r with f, what happens
    • rs + rt
    • This isn't true in general for example: cos(π/6 +π/3) ≠ cos π/6 + cos π/3
  2. Addition formulas for Sine and Cosine
    sin(s + t) =
    sin(s - t) =
    cos(s + t) =
    cos(s - t) =
    • sin(s + t) = (sin s)(cos t) + (cos s)(sin t)
    • sin(s - t) = (sin s)(cos t) - (cos s)(sin t)
    • cos(s + t) = (cos s)(cos t) - (sin s)(sin t)
    • cos(s - t) = (cos s)(cos t) + (sin s)(sin t)
  3. Through the use of addition formulas for sine and cosine we know
    cos (π/2 - θ) =? 
    sin (π/2 - θ) =?
    • sin θ
    • cos θ
  4. In the particular case in the previous function we can say that cosine and sine are cofunctions. (explain)
    • The angles of interest were "π/2 - θ" & "θ" and they both add up to π/2 making them complementary angles.
    • So the function value of one function at a number is equal to the cofunctions's value at the complementary number.
  5. How would you find the cos 15°
    • Turn it into something in which my addition formulas can work with like:
    • cos 15° = cos (45° - 30°)
  6. Addition formulas for tangent:
    tan (s + t) = 
    tan (s - t) =
    • tan (s + t) = (tan s + tan t) / (1 - tan s tan t) 
    • tan (s - t) = (tan s - tan t) / (1 + tan s tan t)
  7. The Double-Angle Formulas
    sin 2θ =
    cos 2θ = 
    tan 2θ =
    • sin 2θ = 2 sin θ cos θ
    • cos 2θ = cos2θ - sin2θ
    • tan 2θ = (2 tan θ) / (1 - tan2θ)
  8. The Half-Angle Formulas *pg 576
    sin s/2 =
    cos s/2 =
    tan s/2 =
    • sin s/2 = ± (√1 - cos s/2)
    • cos s/2 = ± (√1 + cos s/2)
    • tan s/2 = (sin s) / (1+ cos s)
  9. In the half-angle formulas the ± symbol is intended to mean either ______ or _____ not _____. The sign before the radical is determined by the ______ in which the angle (or acr) s/2 terminates
    • positive or negative 
    • both
    • quadrant
  10. How do you find the identities for sin 2 θ, cos 2 θ, and tan 2 θ
    The same way replace 2 θ by (θ + θ) and use the appropriate addition formula
  11. Formulas for cos 3 θ and sin 3 θ
    • cos 3 θ = 4 cos3θ - 3 cos θ
    • sin 3 θ = 3 sin θ - 4 sin3θ
  12. Equivalent forms of the formula: 
    cos 2θ = cos2θ - sin2θ
    • cos 2θ = 2 cos2θ - 1
    • cos 2θ = 1 - 2 sin2θ
    • cos2θ = (1 + cos 2θ)/2
    • sin2θ = (1 - cos 2θ)/2
  13. cos22t =
    (1 + cos 4t)/2
  14. sin (2x) =
    2 cos(x) sin(x)
  15. As indicated in figure 1a (pg 608), the sine function is not ___ ___ ___, therefore there is no _____ _______. The only way around this the _____ _____ function.
    Image Upload
    • one-to-one
    • inverse function
    • restricted sine function *ex y = sin x (-π/2≤ x ≤π/2)
  16. State the domain and range of the restricted sine function has a domain of in the picture. State whether it is one to one, if so or if not, what is the implication?
    Image Upload
    • Domain: [-π/2, π/2]
    • Range: [-1,1]
    • It is one to one, meaning it has an inverse function
  17. What are the two common notations used to denote the inverse sine function:
    • y = sin-1
    • y = arcsin x
  18. State the range and domain, what do you notice?
    Image Upload
    • Domain: [-1,1]
    • Range: [-π/2,π/2]
    • Domain and range are swapped in inverse functions
  19. *Recurring theme: Inverse functions are symmetrical about the line ______
    y = x
  20. sin-1x is that number in the interval [-π/2,π/2] whose sine is ___. With that in mind, what do you do when asked to "evaluate: sin-1(1/2)"?
    • x
    • Think which number WITHIN the (restricted sine function's) interval [-π/2,π/2] has a sine or y-value of 1/2. The answer is π/6
  21. acrsin 1/2
    π/6
  22. Are sin-10 and (1/sin 0) the same? (explain)
    • No, the quantity sin-10 is that number WITHIN the interval [-π/2,π/2] whose sine is 0. Since 0 is in the interval [-π/2,π/2] and sin 0 = 0, we conclude sin-10 = 0
    • On the other hand, since sin 0 = 0, the expression (1/sin 0) is not even defined
  23. f[f-1(x)] = x for every x in the domain of f-1
    f-1[f(x)] = x for every x in the domain of f
    How does this apply to sines and arcsines?
    • sin(sin-1x) = x for every x in the interval [-1,1]
    • sin-1(sinx) = x for every x in the interval [-π/2,π/2]
  24. sin-1(sin 2)
    • Cant be 2 because two is not WITHIN the interval [-π/2,π/2]
    • However, a) π-2 is in that interval & b) sin 2 = sin (π - 2). So the original question is equivalent to asking sin-1[sin(π - 2)] = π - 2
  25. sin(sin-12)
    2 is not in the in the domain of the inverse sine function so that expression is undefined
  26. Review Example 5 pg 611 if there's time
  27. Restricted cosine function (3)
    • y = cos x (0 ≤ x ≤ π)
    • must have domain [0,π] and range [-1,1]
  28. Two ways to denote inverse cosine function
    y = cos-1x or arccos x
  29. arccos domain and range
    Image Upload
    • domain: [-1,1]
    • range: [0,π]
  30. f[f-1(x)] = x for every x in the domain of f-1
    f-1[f(x)] = x for every x in the domain of f
    How does this apply to cosines and arc-cosines?
    • cos(cos-1x) = x for every x in the interval [-1,1]
    • cos-1(cosx) = x for every x in the interval [0,π]
  31. arccos(cos 4)
    Can't be 4, it is not within the restricted interval [0,π]. However, a) 2π - 4 is within the interval & b) cos 4 = cos(2π - 4)
  32. List the 3 key features that define the restricted tangent function:
    Image Upload
    • y = tan x (-π/2< x <π/2)
    • domain: (-π/2,π/2)
    • range: R
  33. Two common notations for the inverse tangent function
    y = tan-1 x or y = arctan x
  34. f[f-1(x)] = x for every x in the domain of f-1
    f-1[f(x)] = x for every x in the domain of f
    How does this apply to tangents and arc-tangents?
    • tan(tan-1x) = x for every real number
    • tan-1(tan x) = x for every x in the open interval (-π/2,π/2)

Card Set Information

Author:
chikeokjr
ID:
333591
Filename:
Analytical Trigonometry
Updated:
2017-08-21 03:30:43
Tags:
Precalculus
Folders:
Precalculus Final Exam
Description:
8.1,8.2 & 8.5
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