Solving Trigonometric Equations: A Step-by-Step Guide
Trigonometric equations can often look intimidating at first glance, but by breaking down the problem and using basic trigonometric identities, we can find the values of unknown variables such as x and y. This guide will walk you through the process of solving the given equations: cos(xy/2) * cos(y/2) 0 and cos(yx/2) * cos(x/2) 0.
Step 1: Simplifying the Equations
Let's start with the first equation:
(1) cos(x * y/2) * cos(y/2) 0
We can split this into two possible cases by setting each factor to zero:
cos(xy/2) 0 or cos(y/2) 0
Similarly, for the second equation:
(2) cos(y * x/2) * cos(x/2) 0
Again, we can split it into two cases:
cos(yx/2) 0 or cos(x/2) 0
Step 2: Using Trigonometric Identities
Let's define:
A x * y/2, B y/2
And rewrite the first equation using cos A * cos B 0:
cos A * cos B cos A * cos B - sin A * sin B
cos A * cos B 0 - sin A * sin B
cos A * cos B -sin A * sin B
Therefore:
cos(x * y/2) * cos(y/2) -sin(x * y/2) * sin(y/2)
Step 3: Comparing and Solving for x and y
From the given equations, we can write:
-sin(x * y/2) * sin(y/2) -sin(y * x/2) * sin(x/2)
This simplifies to:
x * y/2 y * x/2
Which implies:
x y
Substituting y with x in the first equation:
cos(x * x/2) * cos(x/2) 0
This can be further simplified to:
cos(3x/2) * cos(x/2) 0
Breaking this into two cases:
Case 1: cos(3x/2) 0
Solving for x:
3x/2 n * pi
x (2n * pi) / 3
Thus, x y (2n * pi) / 3 where n is an integer.
Case 2: cos(x/2) 0
Solving for x:
x/2 n * pi
x 2n * pi
Thus, x y 2n * pi where n is an integer.
Conclusion:
By systematically solving the given trigonometric equations, we determined the possible values of x and y to be:
Either x y (2n * pi) / 3 Or x y 2n * piFormulas Used:
In the breakdown of the solutions, we used the following identities:
cos(A * B) cos(A) * cos(B) - sin(A) * sin(B) cos(2A) 2cos^2(A) - 1Understanding these identities and their application is crucial for solving such problems efficiently.