For the solutions of 4tantheta = 5tantheta + sqroot3, can both these answers be acceptable?
120 + 180n
-60 + 180n
To solve the equation 4tan(theta) = 5tan(theta) + sqrt(3), we need to isolate the variable theta. Let's go through the steps to find the solutions:
1. Start by subtracting 5tan(theta) from both sides of the equation:
4tan(theta) - 5tan(theta) = sqrt(3)
2. Simplify the left side by combining like terms:
-tan(theta) = sqrt(3)
3. To isolate the tangent function, we need to take the inverse tangent (arctan) of both sides:
arctan(-tan(theta)) = arctan(sqrt(3))
4. The arctan function and the tangent function are inverses of each other, so they cancel out on the left side:
-theta = arctan(sqrt(3))
5. Since we want to find the values of theta, we can multiply both sides by -1 to get positive theta:
theta = -arctan(sqrt(3))
At this point, we have found one solution for theta, but we can also find the general form for the solutions by considering the periodic nature of the tangent function.
The tangent function has a period of 180 degrees (or pi radians). This means that adding or subtracting a multiple of 180 to the angle theta will still satisfy the equation. Thus, the general form for the solutions is:
theta = -arctan(sqrt(3)) + 180n
where n is an integer that can take any value.
Therefore, the two solutions you provided:
- 120 + 180n
- 60 + 180n
are both acceptable, as they represent different values of theta that satisfy the original equation.