How can I get sinx, given 1-cos(x)-cos(y)+sin(x)cos(y)=0, 1+cos(x)-sin(y)+sin(x)sin(y)=0

[Xihe(PRC)]

Yesterday my friend turned to me to discuss something about her homework.When I got this I got stuck at the very beginning.

1 - cos(x) - cos(y) + sin(x)cos(y) = 0
1 + cos(x) - sin(y) + sin(x)sin(y) = 0

How can I get sinx?(We are learning about the relations between sinx,cosx and tanx recently,Such as tanx=sinx/cosx,(sinx^2)+(cosx)^2=1)
Thanks for your help.

 

[Dr.Peterson]

Yesterday my friend turned to me to discuss something about her homework.When I got this I got stuck at the very beginning.

1 - cos(x) - cos(y) + sin(x)cos(y) = 0
1 + cos(x) - sin(y) + sin(x)sin(y) = 0

How can I get sinx?(We are learning about the relations between sinx,cosx and tanx recently,Such as tanx=sinx/cosx,(sinx^2)+(cosx)^2=1)
Thanks for your help.

One approach would be to start by eliminating functions of y. Observe that the first equation can be solved for cos(y) and the second for sin(y). Do so, and use what you know about the two functions to eliminate y entirely.

Then you will have one equation in sin(x) and cos(x), which you can solve for sin(x).

Now please show us whatever you have tried (that way, or any other way) so we can nudge you, if necessary, in a useful direction.

 

[JeffM]

This is a different starting point from Dr. Peterson's, but it leads to the same place.

My starting point when I am faced with something new is to ask myself whether it has any similarity to some class of problems I already understand. If so, I start doing algebraic manipulations that will make this problem look more like problems in that class. So when I saw this problem, I thought immediately about two general kinds of problems: one dealing with sines and cosines of sums of variables and the other dealing with n simultaneous equations in n unknowns. I do not know whether either idea will bear fruit, but I start experimenting.

In this case, as Dr. Peterson has pointed out, the n simultaneous equations idea is very easy to experiment with and quickly leads to an interesting result when you isolate the functions of y on one side of the equations.

The point is that staring at a problem is seldom productive. Try things; one may spark a creative thought.

 

[skeeter]

note the last two terms in the first equation have cos(y) in common
similarly, the last two terms in the second equation have sin(y) in common

try factoring cos(y) and sin(y) respectively from those two last terms, then sum the two resulting equations to eliminate cos(x) …

 

[Dr.Peterson]

the n simultaneous equations idea is very easy to experiment with and quickly leads to an interesting result when you isolate the functions of y on one side of the equations.

This is exactly how I started. In addition, I did something I often do with an equation like this: I abbreviate, for example, sin(x) as sx, so I can think of it as two equations in four unknowns, sx, sy, cx, cy:

1 - cx - cy + sx cy = 0​

1 + cs - sy + sx sy = 0​

This make it more compact (and less scary); I can see what's there more easily. And by using pairs of letters rather than four single-letter variables, I can see which are related by identities.

It's also worth keeping in mind that, although we've only written two equations, which would not be enough to fully solve for four separate unknowns, there are other equations waiting in the wings, such as

cx^2 + sx^2 = 1​