How would I solve this equation for x in the interval 0<x<2pi
tan^2x sinx - sinx/3 = 0
Please explain it step by step..ive been stuck on this question for far too long..sigh
Try factoring out sin(x).
how would i do that?
tan^2(x)*sin(x)-sin(x)/3 = 0
can be rewritten as
sin(x) [ tan^2(x) - 1/3] = 0
which is solvable.
To solve the equation tan^2x sinx - sinx/3 = 0 for x in the interval 0 < x < 2π, you can follow these steps:
Step 1: Factor out sinx from the equation:
sinx(tan^2x - 1/3) = 0
Step 2: Set each factor equal to zero:
sinx = 0 or tan^2x - 1/3 = 0
Step 3: Solve the first factor sinx = 0:
In the given interval (0 < x < 2π), sinx = 0 occurs at x = 0 and x = π.
Step 4: Solve the second factor tan^2x - 1/3 = 0:
Rearrange the equation to isolate tan^2x:
tan^2x = 1/3
Step 5: Take the square root of both sides:
tanx = ±√(1/3)
Step 6: Find the reference angle for tanx = √(1/3):
The reference angle for tanx is the angle whose tangent is √(1/3). This angle can be found using the inverse tangent function:
tan^(-1)(√(1/3))
Using a calculator, the reference angle is approximately 0.615 radians (or approximately 35.26 degrees).
Step 7: Determine the possible values of x:
Since tanx(x) = tan(x + π), the possible values of x are:
x = 0.615, π - 0.615, π + 0.615, 2π - 0.615
Step 8: Check the values of x:
Plug each value of x into the original equation tan^2x sinx - sinx/3 = 0 to ensure they satisfy the equation.
In summary, the solutions to the equation tan^2x sinx - sinx/3 = 0 in the interval 0 < x < 2π are x = 0, π, 0.615, π - 0.615, π + 0.615, and 2π - 0.615.