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INSTRUCTORS Raffi Hovasapian John Zhu
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Lecture Comments (1)

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Post by John Michael Musaazi on September 7, 2014

for example 3 were do you get 3/2

Trigonometric Rules

  • Memorize thoroughly!
  • Careful with negative signs when dealing with cosines
  • Careful not to drop any constants when concentrating on trig portion

Trigonometric Rules

Use the limit definition of a derivative and trig identities to prove that [d/dx] sin(x) = cos(x)
  • f′(x) = limh → 0 [(sin(x + h) − sin(x))/h]
  • = limh → 0 [(sin(x)cos(h) + cos(x)sin(h) − sin(x))/h]
  • = limh → 0 [(sin(x)cos(h) − sin(x))/h] + limh → 0[cos(x)sin(h)/h]
  • = sin(x)limh → 0 [(cos(h) − 1)/h] + cos(x) limh → 0 [sin(h)/h]
  • = sin(x) * 0 + cos(x) * 1 =
cos(x)
Find the derivative of y = sin2(x) + cos2(x)
  • dy = [d/dx] (sin2(x) + cos2(x))
  • = [d/dx] 1 =
0
Prove that [d/dx] cos(x) = −sin(x) using the taylor series expansions of cos(x) and sin(x)
cos(x) = 1 − [(x2)/2!] + [(x4)/4!] − [(x6)/6!] + ...
sin(x) = x − [(x3)/3!] + [(x5)/5!] + ...
[d/dx] cos(x) = [d/dx] (1 − [(x2)/2!] + [(x4)/4!] − [(x6)/6!] + ...)
= − x + [(x3)/3!] − [(x5)/5!] + ...
= − (x − [(x3)/3!] + [(x5)/5!] + ...)
= − sin(x)
Find the derivative of y = 3cos(x) − [7/4]
  • dy = [d/dx] ( 3 cos(x) − [7/4])
  • = 3 [d/dx] cos(x) − [d/dx] [7/4] =
−3 sin(x)
Find the derivative of y = [sin(2x)/cos(x)]
  • dy = [d/dx] [sin(2x)/cos(x)]
  • = [d/dx] [2sin(x)cos(x)/cos(x]
  • = 2 [d/dx]sin(x) =
2 cos(x)
Find the derivative of y = 3sec(x) + x2
  • dy = [d/dx] (3sec(x) + x2) =
3sec(x)tan(x) + 2x
Find the derivative of y = 3tan(x) + 5cot(x)
  • dy = [d/dx] (3tan(x) + 5cot(x))
  • = 3sec2(x) + 5(−csc2(x)) =
3sec2(x) − 5csc2(x)
Find the derivative of y = [tan(x)cos(x)/(sin2(x))]
  • dy = [d/dx] [tan(x)cos(x)/(sin2(x))]
  • = [d/dx] (tan(x) [cos(x)/(sin2(x))])
  • = [d/dx] ([sin(x)/cos(x)] [cos(x)/(sin2(x))])
  • = [d/dx] [1/sin(x)]
  • = [d/dx] csc(x)
= −csc(x)cot(x)
Find the derivative of y = [5/cos(x)]
  • dy = [d/dx] [5/cos(x)]
  • = [d/dx] 5sec(x)
  • = 5 [d/dx] sec(x) =
5 sec(x) tan(x)
Find the derivative of y = 2cos(x) + 3sin(x) + 5sec(x) + 7csc(x) + 11tan(x) + 13cot(x)
  • dy = [d/dx] (2cos(x) + 3sin(x) + 5sec(x) + 7csc(x) + 11tan(x) + 13cot(x))
−2sin(x) + 3cos(x) + 5sec(x)tan(x) − 7csc(x)cot(x) + 11sec2(x) − 13csc2(x)

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Trigonometric Rules

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Trigonometric Rules 0:07
    • COS X
    • Find Derivative
  • Example 1 2:46
  • Example 2: COS Function 3:09
  • Example 3: Composite Expression 3:54
  • Example 4: Sec Function 5:02
  • Example 5: CSC 5:33
  • Example 6L COT 6:42