Solve trigonometric equation by finding all real numbers tanxsin2x=0
To find all real solutions to the trigonometric equation tanxsin2x = 0, we need to find the values of x that make the product of tanx and sin2x equal to zero.
Using the zero-product property, we know that if a product of two factors equals zero, then at least one of the factors must equal zero.
So, either tanx = 0 or sin2x = 0.
To find the solutions for tanx = 0, we need to find the values of x for which the tangent of x equals zero. The tangent of x is equal to zero when x is a multiple of pi radians. Therefore, the solutions for tanx = 0 are x = n*pi, where n is an integer.
To find the solutions for sin2x = 0, we need to find the values of x for which sin2x = 0. We can rewrite sin2x as sin(2x) using the double-angle identity for sine:
sin2x = 2sinx*cosx = 0
Now we have two possibilities: sinx = 0 or cosx = 0.
If sinx = 0, then x = n*pi, where n is an integer.
If cosx = 0, then x = (2n + 1)*pi/2, where n is an integer.
Therefore, the solutions to the equation tanxsin2x = 0 are:
x = n*pi (where n is an integer)
or
x = (2n + 1)*pi/2 (where n is an integer)