A vertical asymptote occurs when the denominator of a fraction becomes zero. This happens when the independent variable (usually x, θ, etc.) has a value which makes the denominator zero. Remember that trig functions are ratios which are essentially fractions (the ratio of two side lengths).
For example y=(x+1)/(x-1) has a vertical asymptote when x=1; y=sinθ has no asymptotes but y=tanθ has asymptotes when cosθ=0 (tanθ=sinθ/cosθ, so cosθ=0 for odd multiples of π/2).
If a denominator is a quadratic or some higher degree polynomial, there are asymptotes for each of the (real) zeroes of the polynomial.