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INSTRUCTORS Raffi Hovasapian John Zhu
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For more information, please see full course syllabus of Calculus AB
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Lecture Comments (2)

0 answers

Post by John Michael Musaazi on August 3, 2014

man get detailed enough when teaching,some formulas we don't know were you got them from.

0 answers

Post by Joshua Spears on February 13, 2013

In example 5, how did you switch from subtraction to addition in the denominator?

Trigonometric Identities

  • Used to manipulate expressions into more “useful” forms
  • Must know very well, otherwise very difficult to use

Trigonometric Identities

Simplify y = x2 + 4x + 4 + cos2 x + sin2 x
  • y = x2 + 4x + 4 + cos2 x + sin2 x
  • = (x + 2)2 + cos2 x + sin2 x
= (x + 2)2 + 1
Simplify y = sin(2x)sec(x)
  • y = sin(2x)sec(x)
  • = [sin(2x)/cos(x)]
  • = [2sin(x)cos(x)/cos(x)]
= 2sinx
Simplify y = sin(x) + cos2(x)csc(x).
  • y = sin(x) + cos2 (x) csc(x)
  • y = sin(x) + [(cos2(x))/sin(x)]
  • sin(x) y = sin2 (x) + cos2 (x)
  • sin(x) y = 1
  • y = [1/sin(x)]
y = cscx
Simplify y = cos(2x) + 2sin2(x)
  • y = cos(2x) + 2sin2(x)
  • = cos(2x) + 2 ([(1 − cos(2x))/2])
  • = cos(2x) + 1 − cos(2x)
= 1
Simplify y = sin(4x)sec2(2x)
  • y = sin(4x)sec2(2x)
  • = [sin(4x)/(cos2(2x))]
  • = [2sin(2x)cos(2x)/(cos2(2x))]
  • = 2[sin(2x)/cos(2x)]
= 2tan(2x)
Simplify y = sin(−x)sec(−x)
  • sin is an odd function, cos and sec are even functions
  • y = sin(−x)sec(−x)
  • = −sin(x)sec(−x)
  • = −sin(x)sec(x)
  • = −[sin(x)/cos(x)]
= −tan(x)
Simplify y = cos(x + π)
  • y = cos(x + π)
  • = cos(x)cos(π) − sin(x)sin(π)
  • = cos(x) * (−1) − sin(x) * 0
= −cos(x)
Simplify y = sin(x + 2 π)
  • y = sin(x + 2 π)
  • = sin(x)cos(2 π) + cos(x)sin(2 π)
  • = sin(x) * 1 + cos(x) * 0
= sin(x)
Simplify y = sin(x − [(π)/2])
  • y = sin(x − [(π)/2])
  • = sin(x)cos(−[(π)/2]) + cos(x)sin(−[(π)/4])
  • = sin(x)cos([(π)/2]) − cos(x)sin([(π)/4])
  • = 0 − cos(x) * 1
=−cos(x)
Simplify y = sin2(x)(cos(2x) − 2cos2(x))
  • y = sin2(x)(cos(2x) − 2cos2(x))
  • = sin2(x)(cos(2x) − 2 ([(1 + cos(2x))/2]))
  • = sin2(x)(cos(2x) − 1 − cos(2x))
  • = sin2(x) * (−1)
= −sin2(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 Identities

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
  • Types of Functions: Trigonometric- Trig Identities 0:07
    • 4 Identities
    • Pythagorean
    • Double Angle
    • Power Reducing
    • Sum or Difference
  • Couple More Identities 1:59
    • Negative Angle
    • Product to Sum
  • Example 1: Prove 3:00
  • Example 2: Simplify Expression 5:02
  • Example 3: Prove 5:56
  • Example 4: Prove 8:02
  • Example 5: Prove With TAN 12:43