Dividing by trigonometric functions

[frozonecom]

Hello. I was doing a (simple) physics problem and stumbled with a mathematical problem.

I was doing a projectile motions problem and I have set up my equation like this:Δx = Vi (cosθ) (t)
270= 25cosθ t
t = 270 / (250cosθ)

And this is where I'm having problems.
I know from my high school trig that doing division by a trig function is invalid since cosθ can be zero. However, I see no other way to express t (time) with other equations I know.

So, my main question is, is what I did a valid move algebraically?
Are there any exceptions to that rule about dividing by trigonometric functions?

Note: I actually saw a yahooanswer post about this but I really want to know physicsforums's say about this. I know this might come as an easy question for some but it really came very confusing for me.
Help would be very appreciated.

 

[Vinay R Hegde]

In the equation 270= 25cosθ t if assume cosθ to be 0, you will get 270=0! This is not possible. ∴cosθ≠0 and hence you can divide cosθ and get an equation for t.

I think you meant
t = 270 / (25cosθ) and not t = 270 / (250cosθ)

 

[ehild]

Hello. I was doing a (simple) physics problem and stumbled with a mathematical problem.

I was doing a projectile motions problem and I have set up my equation like this:


Δx = Vi (cosθ) (t)
270= 25cosθ t
t = 270 / (250cosθ)

And this is where I'm having problems.
I know from my high school trig that doing division by a trig function is invalid since cosθ can be zero. However, I see no other way to express t (time) with other equations I know.

[/QUOTE]

You do not divide with a cosine function, but with a number. Theta is a well defined angle in case of projectile motion, the angle the projectile was thrown at with respect to the horizontal. It must be 90 (-90) degrees to get cosθ=0 and that corresponds throwing it vertically up or down. There is no horizontal displacement in these cases.

ehild

 

[HallsofIvy]

Hello. I was doing a (simple) physics problem and stumbled with a mathematical problem.

I was doing a projectile motions problem and I have set up my equation like this:Δx = Vi (cosθ) (t)
270= 25cosθ t
t = 270 / (250cosθ)

And this is where I'm having problems.
I know from my high school trig that doing division by a trig function is invalid since cosθ can be zero. However, I see no other way to express t (time) with other equations I know.

$cos(\theta)$ can be 0 but you should be able to realize that in a problem like this, it won't be! $cos(theta)= 0$ for $\theta$ 90 degrees or 270 degrees. I presume your $\theta$ is the angle the trajectory makes with the horizontal so it can't be 270 degrees and 90 degrees means the projective is 'fired' straight up. And, of course, if the projective goes straight up, it will come straight back down- so you could do that case separately. There is no objection to saying "Clearly, if $\theta$ cannot be 90 because then the ball will not have ANY horizontal motion and x can never be 270. If $\theta$ is not 90 degrees, $cos(\theta)$ is not 0 so we can divide by [itex]cos(\theta)".

(I do have an objection to the "25" in one equation becoming "250" in the next! May we assume that extra "0" is a typo?)

So, my main question is, is what I did a valid move algebraically?
Are there any exceptions to that rule about dividing by trigonometric functions?

Note: I actually saw a yahooanswer post about this but I really want to know physicsforums's say about this. I know this might come as an easy question for some but it really came very confusing for me.
Help would be very appreciated.

 

[CellCoree]

Hello,

Thank you for reaching out to us with your question. It is understandable that you are confused about dividing by trigonometric functions in this case.

In general, dividing by a trigonometric function is not a valid algebraic move. As you mentioned, this is because the trigonometric function can be equal to zero, which would make the division undefined.

However, there are some exceptions to this rule. For example, if the trigonometric function is squared, then it can be divided by. In your case, if you have a trigonometric function squared, you can take the square root of both sides of the equation to eliminate the squared function and then divide.

Another exception is if the trigonometric function is part of a larger expression, and dividing by it would not cause the overall expression to be undefined. In your case, you have a constant (270) on one side and a trigonometric function (cosθ) multiplied by a variable (t) on the other side. Dividing by cosθ would not make the overall expression undefined, so it would be a valid algebraic move.

In summary, dividing by a trigonometric function is generally not a valid algebraic move, but there are exceptions to this rule. In your case, it would be valid to divide by cosθ since it is part of a larger expression and would not make the overall expression undefined. I hope this helps clarify your confusion. If you have any further questions, please let me know.