What would be the 2 solutions to this equation?
sinx(tanx) + tanx = sqrt(3)(sinx) + sqrt(3)
I know that "x = pi/3 +pi k" is one, but I need help finding the other.
Cheers
sinx tanx + tanx = √3 sinx + √3
tanx(sinx+1) = √3 (sinx+1)
(tanx-√3)(sinx+1) = 0
tanx = √3
sinx = -1
That help?
To find the solution to the equation sinx(tanx) + tanx = sqrt(3)(sinx) + sqrt(3), you can follow these steps:
1. Rewrite the equation:
sinx(tanx) + tanx - sqrt(3)(sinx) - sqrt(3) = 0
2. Combine like terms:
tanx(sinx + 1) - sqrt(3)(sinx + 1) = 0
3. Factor out (sinx + 1):
(sinx + 1)(tanx - sqrt(3)) = 0
4. Set each factor equal to zero:
sinx + 1 = 0 --> sinx = -1
tanx - sqrt(3) = 0 --> tanx = sqrt(3)
5. Solve for x:
For sinx = -1, the solutions are x = (2n+1)π - π/2, where n is an integer.
For tanx = sqrt(3), the solution is x = π/3 + nπ, where n is an integer.
So the two solutions to the equation are:
1. x = (2n+1)π - π/2 (where n is an integer)
2. x = π/3 + nπ (where n is an integer)
Please note that the solution you mentioned, x = π/3 + kπ, is a generalized form of the second solution.