B. Trigonometric Functions · Radian Measure and Trigonometric Functions

Graphs of the Reciprocal Functions: csc x, sec x and cot x

B2.3About 70 minutes

Learning goals

I can sketch y = csc x, y = sec x and y = cot x from the graphs of sin x, cos x and tan x.
I can explain where each reciprocal function has vertical asymptotes and invariant points.
I can state the domain, range, period and asymptotes of each reciprocal function, with radian measures.
I can compare a function with its reciprocal (sign, size, increasing or decreasing).

Words to know

reciprocal function
For a function f, the function y = 1/f(x). Its values are the reciprocals of the values of f.

invariant point
A point that stays the same under a change. For y = 1/f(x), these are where f(x) = 1 or f(x) = −1, because 1/1 = 1 and 1/(−1) = −1.

cosecant function
y = csc x = 1/sin x.

secant function
y = sec x = 1/cos x.

cotangent function
y = cot x = 1/tan x = cos x/sin x.

local maximum / local minimum
A point higher (or lower) than all nearby points on the graph, though not necessarily the highest (or lowest) overall.

Graphing a reciprocal

You already know three graphs: sin x, cos x and tan x. Their reciprocals can be sketched from those graphs using a few rules about the reciprocal y = 1/f(x):

Zeros become asymptotes. Where f(x) = 0, 1/f(x) is undefined, so there is a vertical asymptote.
±1 stay put. Where f(x) = 1 or −1, the reciprocal has the same value: these are invariant points.
Small becomes large, large becomes small. Where |f(x)| is close to 0, |1/f(x)| is huge.
Same sign. The reciprocal of a positive number is positive; of a negative number, negative.
Opposite direction. Where f is increasing (and not 0), 1/f is decreasing, and the other way round.

y = csc x

Start with y = sin x (dashed below). Its zeros, x = nπ, become vertical asymptotes. At its maximum (π/2, 1) and minimum (3π/2, −1), the reciprocal touches the sine curve. Between those points the cosecant curve bends away from the x-axis, forming U-shapes that open up above the humps and down below the troughs.

y = csc x (solid) drawn from y = sin x (dashed). The asymptotes are at the zeros of sine, and the curves touch only at the invariant points where sin x = ±1.

y = sec x

Do the same with y = cos x. Its zeros, x = π/2 + nπ, become asymptotes, and the U-shapes sit on the maximum and minimum points of cosine, such as (0, 1) and (π, −1). The secant graph is the cosecant graph shifted π/2 to the left, just as cosine is sine shifted π/2 to the left.

y = sec x (solid) drawn from y = cos x (dashed). The asymptotes are at x = π/2 + nπ, where cos x = 0.

y = cot x

The reciprocal of tan x swaps its zeros and asymptotes:

where tan x = 0 (x = nπ), cot x has vertical asymptotes;
where tan x is undefined (x = π/2 + nπ), cot x = cos x / sin x = 0, so cot x has zeros.

The invariant points are where tan x = ±1, such as (π/4, 1) and (3π/4, −1). Since tan x increases on each branch, cot x decreases on each branch. Its period is π, like tangent.

y = cot x (solid) and y = tan x (dashed). The asymptotes of cot x are at the zeros of tan x, and the zeros of cot x are where tan x has asymptotes. Each branch of cot x falls from +∞ to −∞.

	y = csc x	y = sec x	y = cot x
Domain	x ≠ nπ	x ≠ π/2 + nπ	x ≠ nπ
Range	y ≤ −1 or y ≥ 1	y ≤ −1 or y ≥ 1	all real numbers
Period	2π	2π	π
Vertical asymptotes	x = nπ	x = π/2 + nπ	x = nπ
Zeros	none	none	x = π/2 + nπ
Local min / max points	(π/2 + 2nπ, 1) / (3π/2 + 2nπ, −1)	(2nπ, 1) / (π + 2nπ, −1)	none

Properties of the reciprocal trigonometric functions, with n ∈ ℤ and x ∈ ℝ.

Worked example

Example 1: Sketch csc x from sin x

Sketch y = csc x for 0 ≤ x ≤ 2π. Give the asymptotes, the local maximum and minimum points, the domain and the range on this interval.

Sketch y = sin x lightly. Its zeros are 0, π and 2π: draw dashed vertical asymptotes there.
Mark the invariant points (π/2, 1) and (3π/2, −1).
Between 0 and π, sin x is positive and at most 1, so csc x ≥ 1: draw a U opening upward through (π/2, 1), rising toward both asymptotes.
Between π and 2π, sin x is negative, so csc x ≤ −1: draw an upside-down U through (3π/2, −1).

Asymptotes x = 0, π, 2π. Local minimum (π/2, 1); local maximum (3π/2, −1). Domain 0 < x < 2π, x ≠ π. Range y ≤ −1 or y ≥ 1.

Worked example

Example 2: Values near an asymptote

Use a calculator to find csc 0.1, csc 0.01 and csc(−0.01). What do they show about the graph near x = 0?

csc 0.1 = 1/sin 0.1 ≈ 1/0.0998 ≈ 10.02.
csc 0.01 = 1/sin 0.01 ≈ 100.0.
csc(−0.01) ≈ −100.0.

As x → 0 from the right, csc x → +∞; from the left, csc x → −∞. The y-axis (x = 0) is a vertical asymptote.

Worked example

Example 3: Increasing and decreasing

On 0 ≤ x ≤ 2π, where is y = sec x increasing?

sec x increases where cos x decreases, except where cos x = 0.
cos x decreases on [0, π].
Remove x = π/2, where sec x is undefined.

0 ≤ x < π/2 and π/2 < x ≤ π

Worked example

Example 4: The zeros of cot x

Find the zeros and the asymptotes of y = cot x for 0 ≤ x ≤ 2π.

cot x = cos x / sin x. It is 0 where cos x = 0 (and sin x ≠ 0): x = π/2 and 3π/2.
It is undefined where sin x = 0: x = 0, π, 2π.

Zeros π/2, 3π/2; asymptotes x = 0, x = π, x = 2π.

Just for fun

A function and its reciprocal never disagree about signs, and they always meet at 1 and −1. They just have very different opinions about zero.

Hands-on activity

Build csc x from a table

You need: graph paper (or a sheet with a grid drawn in pencil), a ruler, a pencil and a calculator in radian mode.

Make a table with x = 0.1, 0.3, 0.5, 0.8, 1.2, π/2 ≈ 1.57, 2.0, 2.4, 2.8, 3.0 and 3.1.
For each x, record sin x and then csc x = 1 ÷ sin x (two decimals).
Set up axes with 1 unit = 2 cm on the x-axis (to about 3.2) and 1 unit = 1 cm on the y-axis (to about 12; csc 3.1 is about 24, so mark that point with an arrow off the top of the grid). Plot both sin x and csc x.
Draw the dashed asymptotes x = 0 and x = π ≈ 3.14.
Write down the smallest csc value you found and where it happened.

What to look for: csc 0.1 ≈ 10.02 and csc 3.1 ≈ 24.05 (both near asymptotes); the smallest value is csc(π/2) = 1, the invariant point where the two curves touch. The csc curve is a U-shape above the sine hump and never goes below 1.

Your turn

Write the equations of the asymptotes of y = csc x for 0 ≤ x ≤ 2π.

State the range of y = sec x.

What is the period of y = cot x?

Find csc(π/2). What kind of point is (π/2, csc(π/2)) on the graph?

State the domain of y = sec x.

List the zeros of y = cot x for 0 ≤ x ≤ 2π.

Explain why y = csc x has no zeros.

More practice: warm-up

Complete: csc x = 1 ÷ ___.

sec 0 = ?

cot(π/4) = ?

Is y = sec x defined at x = π/2?

What is the period of y = csc x?

More practice: on your own

Find the local maximum and minimum points of y = sec x for 0 ≤ x ≤ 2π.

On 0 ≤ x ≤ 2π, where is y = csc x increasing?

Find csc 3.1 with a calculator and explain the size of the answer.

Explain why y = sec x has no values between −1 and 1.

Find the points where y = cot x and y = tan x intersect for 0 < x < π.

Challenge

Explain why sec x = csc(x + π/2), and describe the graph of y = sec x as a translation of y = csc x.

Is y = csc x even, odd or neither? What about y = sec x? Explain.

Find the equations of the asymptotes and the period of y = sec(2x).

Check your understanding

Quiz

Where does y = csc x have vertical asymptotes?

What is the range of y = sec x?

What is the period of y = cot x?

On 0 < x < π, y = csc x has a local minimum at...

At every x where sin x = 0, the graph of y = csc x has...

Which function is decreasing on every interval between its asymptotes?

For parents and teachers

Big idea (B2.3): csc x, sec x and cot x are reciprocals of sin x, cos x and tan x. Zeros of the original become vertical asymptotes, values of ±1 stay put, small values become large, and the sign never changes.

How to help: Ask your teen to sketch sin x first and then turn it into csc x, saying the five reciprocal rules out loud. Check the sketch at x = π/2 (both equal 1).

Common difficulties:

Thinking csc x is sin⁻¹x (the inverse sine). It is 1/sin x.
Putting secant's asymptotes at the zeros of sine instead of cosine.
Drawing U-shapes that cross y = 1 or y = −1 into the gap.
Mixing up which of csc and sec is paired with sin (each pair has exactly one "co": cosecant–sine, secant–cosine, cotangent–tangent).