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

Sduibek

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

LCKurtz

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]

Sduibek

Okay, thank you!

Curious3141

(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.