Solving a trigonometry equation simultaneously in two variables

[PhysicsKid703]

Homework Statement

Solve the equations sin(x+2y)=1/2 , cos(2x-y)=1/ sqrt(2)

Homework Equations

The Attempt at a Solution

I tried getting a generic solution for both the first and second equation. How do I further proceed? By simultaneously solving? I so, how?

2x-y=2m*pi + or - pi/4

x+2y=n*pi + (-1)^n * pi/6

 

[HallsofIvy]

Yes, solve those two equations simultaneously. For example, if you were to multiply the first equation by 2, getting 4x- 2y= 3m\pi\pm \pi/2. Now, adding that to the second equation eliminates "y" from the equation.

 

[Ray Vickson]

Homework Statement

Solve the equations sin(x+2y)=1/2 , cos(2x-y)=1/ sqrt(2)

Homework Equations

The Attempt at a Solution

I tried getting a generic solution for both the first and second equation. How do I further proceed? By simultaneously solving? I so, how?

2x-y=2m*pi + or - pi/4

x+2y=n*pi + (-1)^n * pi/6

You seem to be saying you do not know how to solve the two equations
2x - y = a\\x + 2y = b
for $x, y$ in terms of $a,b$. Is that really true, or have you just not 'recognized' the problem correctly?

 

[PhysicsKid703]

I'm in 11th grade, doing IIT pre-prep as extra classes.
I do know how to solve simultaneous equations :P. However, I was wondering whether the +- part would affect solving simultaneous equations, because I want a general solution for x and y e.g. n*pi + pi/4 or something.

 

[HallsofIvy]

Well, yes, it will 'affect' the answer but the way the answer is affected is just basic algebra. Have you done as I suggested?

 

[physicspankaj]

i am not sure but i think we can solve it
as
(x+2y)=sin(inverse)1/2=30(degree)
2x-y=cos(inverse)1/sqrt2=90(degree)
amusing 1 and 2 equation
multiply 2 equation by 2
then we will get x=24(degree)
and y=3(degree)

 

[Simon Bridge]

Welcome to PF;

i am not sure but i think we can solve it
as
(x+2y)=sin(inverse)1/2=30(degree)

Well spotted.
The thing is that sin(a)=1/2, not just for for a=30°, but also for a=150° and many other angles besides. In degrees, the full equation goes like:

$a = 180n + 30\times (-1)^n: n=0,1,2,3,\cdots$

This is what is in post #1, with a=x+2y (right at the bottom) - except that PhysicsKid703 is using radians instead of degrees.

 

[PhysicsKid703]

Hmm.
Ok HallsofIvy, I understood what you said and I got the answer written in the back of the book. Thanks so much for the help.
SimonsBridge- Yes, that's the general equation for all the values of theta when sin theta = sin alpha. Thats in fact exactly how it appears as a formula in my text.

 

[Simon Bridge]

Did you understand where that relation came from?
I was thinking that you could use the same sort of observation to get rid of the $\pm$ for the cosine part. Just look at the series of values of "a" that make cos(a)=1/√2 true and try to express them using the (-1)^n style of notation. (hint: 2n-1 is always an odd number for n=1,2,3...).

But you managed to get the "correct" answer anyway so... well done :)