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Are undefined constants valid in trig equations when other constants present?

  1. Jul 22, 2012 #1
    For example:

    tan(pi/2 - pi/4) = (tan(pi/2) - tan(pi/4) ) / 1 + tan(pi/2)tan(pi/4)

    Which of course comes out to:

    undefined + 1 / 1 + undefined

    Does that equal 1, or equal Undefined/No Solution?


    sorry for the poor formatting, I couldn't find the mathprint symbols for pi and fractions
     
  2. jcsd
  3. Jul 22, 2012 #2

    LCKurtz

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    Trig identities are only valid if the various quantities are defined. So, for example, the formula$$
    \tan(a-b) = \frac {\tan a - \tan b}{1 + \tan a \tan b}$$does not apply if either of the angles or their difference is an odd multiple of ##\pi/2##. [Edited, thanks for catching that Curious3141]
     
    Last edited: Jul 22, 2012
  4. Jul 22, 2012 #3
    Okay, thank you!
     
  5. Jul 22, 2012 #4

    Curious3141

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    (Just to correct a small error, the expression *is* perfectly valid if either or both of the angles is an even multiple of [itex]\frac{\pi}{2}[/itex]).

    Whilst this is strictly true (for odd multiples of [itex]\frac{\pi}{2}[/itex]), one can actually use the angle sum identities to calculate the limiting values of expressions.

    For example, we know that [itex]\tan (\frac{\pi}{2} - x) = \cot x[/itex]

    We can prove this easily with a right triangle, but we can also view this as the limit of this expression:

    [itex]\lim_{y \rightarrow \frac{\pi}{2}} \tan (y - x) = \lim_{y \rightarrow \frac{\pi}{2}} \frac{\tan y - \tan x}{1 + \tan x\tan y}[/itex] where [itex]x \neq \frac{(2n+1)\pi}{2}[/itex].

    Since approaching the limit, [itex]|\tan y| >> |\tan x|[/itex], the limit becomes:

    [itex]\lim_{y \rightarrow \frac{\pi}{2}} \frac{\tan y}{\tan x\tan y} = \frac{1}{\tan x} = \cot x[/itex], as expected.

    Usually, when I do rough work "informally", I don't bother with the limits, I just put the values as [itex]\tan \frac{\pi}{2}[/itex] directly. But of course, I'm aware this is mathematically incorrect, just a convenient shorthand.
     
    Last edited: Jul 22, 2012
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