solve the equation in the interval [0,2pi) tan (t/2)-sint=0
tan(t/2) - sint = 0
(1-cost)/sint = sint
1-cost = sin^2 t
1-cost = 1-cos^2 t
cost(1-cost) = 0
cost = 0 or 1
t = pi/2,pi,3pi/2
see
http://www.wolframalpha.com/input/?i=solve+tan%28t%2F2%29+%3D+sint+for+t%3D0..2pi
To solve the equation tan(t/2) - sin(t) = 0 over the interval [0, 2π), we can follow these steps:
Step 1: Recognize the equation and rewrite it.
tan(t/2) - sin(t) = 0
Step 2: Combine the terms involving t by using the double-angle formula for tangent.
tan(t/2) = 2sin(t)cos(t/2)
Step 3: Substitute the double-angle formula in the equation.
2sin(t)cos(t/2) - sin(t) = 0
Step 4: Factor out sin(t) from the equation.
sin(t) * (2cos(t/2) - 1) = 0
Step 5: Set each factor equal to zero and solve for t.
sin(t) = 0 => t = 0, π, or 2π (from the interval [0, 2π))
2cos(t/2) - 1 = 0 => cos(t/2) = 1/2 (add 1 and divide by 2)
=> t/2 = π/3 or t/2 = 5π/3
=> t = 2π/3 or t = 10π/3 (multiply by 2)
Therefore, the solutions for the equation over the interval [0, 2π) are:
t = 0, π, 2π/3, or 10π/3.