Sketch Trigonometric Functions  secant and cosecant
The sketching of the secant and cosecant functions of the form y = a sec( k ( x  d)) and y = a csc( k ( x  d)) are discussed with detailed examples.
Graphing Parameters of y = sec(x) and y = csc(x)
range: (∞ , 1) ∪ (1 , +∞)
Period = 2π
Horizontal Shift (translation) = d , to the left if ( d) is positive and to the right if ( d) is negative.
Vertical asymptotes of y = sec(x) = 1 / cos(x) at the zeros of cos(x) given by x = π/2 + kπ , k = 0 , ~+mn~1, ~+mn~2, ...
Vertical asymptotes of y = csc(x) = 1 / sin(x) at the zeros of sin(x) given by x = kπ , k = 0 , ~+mn~1, ~+mn~2, ...
We need to know how to sketch basic secant and cosecant functions using the identities y = sec(x) = 1 / cos(x) and y = csc(x) = 1 / sin(x) to understand the vertical asymptotes.
y = sec(x) = 1 / cos(x)
All zeros of cos(x) (which is in the denominator) are vertical asymptotes of the sec(x).
y = csc(x) = 1 / sin(x)
All zeros of sin(x) (which is in the denominator) are vertical asymptotes of the csc(x).
Sketching secant and cosecant Functions: Examples with Detailed Solutions

Sketch the graph of y = sec(2x  π/3) over one period.
Solution
Graphing Parameters
range: (∞ ,  1) ∪ (1, +∞)
Period = 2π/2 = π
Vertical asymptotes given by the soltuion to the equation: 2x  π/3 = π/2 + kπ which gives: x = 5π/12 + kπ/2, , k = 0 , ~+mn~1, ~+mn~2, ...
Horizontal Shift: Because of the term  π/3, the graph is shifted horizontally. We first rewrite the given function as: y = sec[2(x  π/6)] and we can now write the shift as being equal to π/6 to the right.
We sketch y = sec(2x  π/3) translating the graph of y = sec(2x) by π/6 to the right (red graph below) so that the sketched period starts at π/6 and ends at π/6 + π = 7π/6 which is one period equal to π.

Sketch the graph of y =  3 csc(x/2 + π/2) over one period.
Solution
Graphing Parameters
range: (∞ , 3) ∪ (3, +∞)
Period = 2π/k = 2 π / (1/2) = 4 π
Vertical asymptotes given by the solution to the equation: x/2 + π/2 = kπ which gives: x = (2k1)π, , k = 0 , ~+mn~1, ~+mn~2, ...
Horizontal Shift: Because of the term π/2, the graph is shifted horizontally. We first rewrite the given function as: y =  3 csc[(1/2)(x + π)] and we can now write the shift as being equal to π to the left.
We sketch  3 csc(x/2 + π/2) by translating the graph of y =  3 csc(x/2) to the left by π (red graph below) so that the sketched period starts at π and ends at π + 4 π = 3π which is an interval equal to one period.


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