Solve the given triginometric equation analytically
2 sinx = tanx
2sinx - sinx/cosx = 0
(2sinxcosx - sinx)/cosx = 0 , multiply by cosx
2sinxcosx - sinx = 0
sinx(2cosx - 1) = 0
so sinx = 0 or cosx = 1/2
I am sure you can finish it from here.
2sinxcosx+cosx=0
To solve the given trigonometric equation analytically, we need to simplify both sides of the equation and find the values of x that satisfy the equation.
Starting with the equation:
2 sin(x) = tan(x)
We know the identity that relates tangent and sine: tan(x) = sin(x) / cos(x)
So we can rewrite the equation as:
2 sin(x) = sin(x) / cos(x)
Next, we can multiply both sides of the equation by cos(x) to eliminate the denominator:
2 sin(x) * cos(x) = sin(x)
Applying the double-angle identity for sine: sin(2x) = 2sin(x)cos(x), we can rewrite the equation as:
sin(2x) = sin(x)
Now we have an equation involving only sine. By using the identity sin(a) = sin(b), the solutions will be:
2x = x + 2nπ where n is an integer representing the solution set
or
2x = π - x + 2nπ where n is an integer representing the solution set
1) Solving for 2x = x + 2nπ:
Subtracting x from both sides of the equation:
x = 2nπ
Now we have the solution x = 2nπ, where n is an integer.
2) Solving for 2x = π - x + 2nπ:
Adding x to both sides of the equation:
3x = π + 2nπ
Dividing both sides by 3:
x = (π + 2nπ) / 3
Simplifying the expression:
x = (π(1 + 2n)) / 3
Now we have the solution x = (π(1 + 2n)) / 3, where n is an integer.
So, the solutions to the equation 2 sin(x) = tan(x) are:
x = 2nπ, where n is an integer
x = (π(1 + 2n)) / 3, where n is an integer.