1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

HELP with trigonometric identites

  1. Dec 6, 2014 #1
    Okay so I'm solving this equation: 2sin(x)-3cos(x)=1
    I'm solving it by equating the left hand side to a single trigonometric equation.
    What I did first: I equated the left hand side to a sine function. So
    2sin(x)-3cos(x)=Rsin(x+a). Then I used the addition formula.
    Rsin(x+a)= Rsin(x)*cos(a)+Rcos(x)*sin(a). Then I equated the coefficients.
    So, Rcos(a)=2 and Rsin(a)=-3.
    So I solved for R & a (not gonna go into the details) and found the R=sqrt(13) and a=-56.31. So 2sin(x)-3cos(x)=sqrt(13)sin(x-56.31)
    Then I said sqrt(13)sin(x-56.31)=1 (as required in first equation), solved for x and got 72.4 and 220.2 (if 0<x<360). This, according to the text book solution, is the correct answer.
    HERE COMES THE PROBLEM
    As stated, I equated the left hand side to Rsin(x+a). But surely, it doesn't matter whether I equate the equation to Rsin(x+a), Rsin(x-a), Rcos(x+a) or Rcos(x-a). Surely, when I solve for x, it should be the same answer, regardless which formula I used. So I tested this out.
    I equated 2sin(x)-3cos(x) to Rcos(x+a) instead
    Expanded Rcos(x+a) to Rcos(x)cos(a)-Rsin(x)sin(a)
    Equated coefficients: Rcos(a)=-3 & -Rsin(a)=2, so Rsin(a)=-2
    Solved for R and a, and got R=sqrt(13) and a=33.69
    So 2sin(x)-3cos(x)=sqrt(13)cos(x+33.69).
    Set the equation: sqrt(13)cos(x+33.69)=1
    Solved for x and got, *drum roll*, 40.21 and 252.41. Two completely different answers!!!!!!
    Got so frustrated that I decided to generate the graphs on a graph sketcher to see what was going on. Turns out that sqrt(13)sin(x-56.31) generates the same graph as 2sin(x)-3cos(x), thus confirming that it is a correct equation.
    However, the graph of sqrt(13)cos(x+36.69) generates the REFLECTION in the x axis, of 2sin(x)-3cos(x).
    Why is this happening!!! What have I done wrong in calculating the cos equation to mean that it doesn't produce the same graph as the others!!!
     
  2. jcsd
  3. Dec 6, 2014 #2

    Mark44

    Staff: Mentor

    cos(x) = -sin(x - 90°), which you can verify by expanding the right side.

    So cos(x + 33.69°) = -sin(x - 90° + 33.69°) = -sin(x + 56.31°)
    So the left and right sides are equal in magnitude, but opposite in sign. Adding a sign to sin(x + 56.31°) causes a reflection across the x-axis.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: HELP with trigonometric identites
  1. Trigonometric equation (Replies: 2)

Loading...