Determine the total number of solutions for the given equation: tan(2θ-pi/2)=1 on the interval [0, 2pi]
sin (2θ-pi/2) / cos (2θ-pi/2) = 1
sin (2θ-pi/2) = cos (2θ-pi/2)
that happens when we are 45 degrees from any axis in quadrants 1 or 3
(2θ-pi/2) = pi/4
2 θ = 3 pi/4
θ = 3 pi/8
(2θ-pi/2) = 5 pi/4
2θ = 7 pi/4
θ = 7 pi/8
(2θ-pi/2)= 2 pi + pi/4 = 11 pi/4
θ =11 pi/8
continue until θ is > 2 pi
(2θ-pi/2) = 3 pi + pi/4
To determine the total number of solutions for the given equation, we need to find the values of θ that satisfy the equation tan(2θ - π/2) = 1 on the interval [0, 2π].
Step 1: Rewrite the equation using a trigonometric identity.
By using the identity tan(π/4) = 1, we can rewrite the equation as tan(2θ - π/2) = tan(π/4).
Step 2: Convert the equation to an equation of the form tan(x) = tan(y).
Using the identity tan(A - B) = (tan(A) - tan(B))/(1 + tan(A) * tan(B)), we can rewrite the equation as:
(tan(2θ) - tan(π/2))/(1 + tan(2θ) * tan(π/2)) = tan(π/4).
Step 3: Simplify the equation.
Since tan(π/2) is undefined, we can rewrite the equation as:
(tan(2θ) - undefined)/(1 + tan(2θ) * undefined) = 1.
Based on this equation, we can conclude that there are no solutions for θ in the interval [0, 2π]. This is because the equation leads to an undefined expression when tan(2θ) = undefined.
Therefore, the total number of solutions for the given equation is 0.