Thread: Find the value R

Given that the two equations:
R sin⁡〖60°〗=10 sin⁡〖θ°〗+v
R cos⁡〖60°〗=10 cos⁡〖θ°〗

How do I get the value R ? Please show me how to do it. Thanks.

Is known.

Originally Posted by alexander9408

Given that the two equations:
R sin⁡〖60°〗=10 sin⁡〖θ°〗+v
R cos⁡〖60°〗=10 cos⁡〖θ°〗

How do I get the value R ? Please show me how to do it. Thanks.

I assume is known.



This can be solved for

both are unknown

Well then there are three unknowns and two equations. This cannot be solved for a unique solution.

well then, can you form this equation θ°=60° - sin^(-1)⁡〖v/20〗using the two equations above?

Originally Posted by alexander9408

well then, can you form this equation θ°=60° - sin^(-1)⁡〖v/20〗using the two equations above?

Yes this can be done by eliminating from the above two equations?

Spoiler:

Dividing the given equations we have





NOTE: Before division by you need to argue that you are not dividing by zero first.

Originally Posted by kalyanram

NOTE: Before division by you need to argue that you are not dividing by zero first.
[/SPOILER]

What do you mean by this?

Originally Posted by alexander9408

What do you mean by this?

Have you considered the case when .

By the way the general solution is . (why?)

Originally Posted by kalyanram

Have you considered the case when .

By the way the general solution is . (why?)

Since the question have been given that the v<10 , when I substitute v=10 to the equation, then I will get θ = 30° , so the θ won't exceed 30°.
The value of sin is positive at first and second quadrant, and negative at third and fourth quadrant, is that so?

Originally Posted by alexander9408

Since the question have been given that the v<10

When we have and which satisfies the condition .

Anyway the point I wanted to highlight was not that. I am probably confusing you more that I intend to so here is how I would like you to analyze.

Step1: There are three unknown , and two equations so if you want to relate any two you have to eliminate the other.
Relating by eliminating was shown above. Now here the discriminant of the quadratic should be for the range of values specified on this gives us an additional criteria that . So permissible values of are .

Step 2: Relating and
While eliminating any variable by trying to divide two expressions (like I did) you have to be careful as to the expression in the denominator is . This is the case when . So you can deal with it as a separate case and argue it out, or try to avoid division if possible. And in this case you can avoid division of the two expressions as shown below:
From the first equation w
We have ,

from here we again get

which has been solved above. The point to be noted here is that eliminating as shown here we have avoided any possibility of division by .

Step 3: General values of .

Originally Posted by alexander9408

when I substitute v=10 to the equation, then I will get θ = 30° , so the θ won't exceed 30°. The value of sin is positive at first and second quadrant, and negative at third and fourth quadrant, is that so?

As I have mentioned above when you obtain a solution for a variable it is all the more important to verify if the solution is valid. When solving for Step 2, we obtain as the principle values of the angle. As stated by you we already know that and also so we need all possible solutions of we obtain the form . Now we need to see if this solution is a valid solution for that we need (why?) which is also consistent with what we got in the solution for .