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]

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

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]

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 a multiple of $\pi/2$.

Okay, thank you!

 

[Curious3141]

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 a multiple of $\pi/2$.

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

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

For example, we know that $\tan (\frac{\pi}{2} - x) = \cot x$

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

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

Since approaching the limit, $|\tan y| >> |\tan x|$, the limit becomes:

$\lim_{y \rightarrow \frac{\pi}{2}} \frac{\tan y}{\tan x\tan y} = \frac{1}{\tan x} = \cot x$, as expected.

Usually, when I do rough work "informally", I don't bother with the limits, I just put the values as $\tan \frac{\pi}{2}$ directly. But of course, I'm aware this is mathematically incorrect, just a convenient shorthand.

 

[CellCoree]

In trigonometric equations, undefined constants are not considered valid. In the given example, the expression on the left side of the equation, tan(pi/2 - pi/4), is undefined because the tangent function is not defined at pi/2 - pi/4. This means that the equation as a whole does not have a valid solution. While it may appear that the right side of the equation simplifies to 1, this is not a valid solution as it is obtained through an invalid operation. Therefore, the equation has no solution. It is important to note that undefined constants should not be treated as regular numbers in mathematical equations. Instead, they should be handled with caution and not be included in calculations.