# Graph 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 and Graphing secant and cosecant Functions: Examples with Detailed Solutions

### Example 1

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. ### Example 2

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 = (2k-1)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. 