Secant, Cosecant, and Cotangent (HSC SSCE Mathematics Advanced): Revision Notes

Secant, Cosecant, and Cotangent

Introduction to reciprocal trigonometric ratios

In addition to the three basic trigonometric ratios (sine, cosine, and tangent), there are three reciprocal trigonometric ratios that extend our toolkit for working with right-angled triangles. These are secant (sec), cosecant (cosec), and cotangent (cot).

Each reciprocal ratio is defined as the multiplicative inverse of one of the basic ratios. Understanding these ratios allows you to work more efficiently with certain trigonometric problems and prepares you for more advanced mathematical concepts.

The three reciprocal ratios

To understand these ratios, we first need to recall the sides of a right-angled triangle relative to an angle $\theta$.

Secant (sec θ)

The secant of an angle is the reciprocal of its cosine.

$\sec \theta = \frac{1}{\cos \theta}$

In terms of the sides of a right-angled triangle, secant is the ratio of the hypotenuse to the adjacent side.

$\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}}$

where $0 < \theta < 90°$.

This ratio tells us how many times longer the hypotenuse is compared to the adjacent side.

Cosecant (cosec θ)

The cosecant of an angle is the reciprocal of its sine.

$\operatorname{cosec} \theta = \frac{1}{\sin \theta}$

In terms of the sides of a right-angled triangle, cosecant is the ratio of the hypotenuse to the opposite side.

$\operatorname{cosec} \theta = \frac{\text{Hypotenuse}}{\text{Opposite}}$

where $0 < \theta < 90°$.

This ratio tells us how many times longer the hypotenuse is compared to the opposite side.

Cotangent (cot θ)

The cotangent of an angle is the reciprocal of its tangent.

$\cot \theta = \frac{1}{\tan \theta}$

In terms of the sides of a right-angled triangle, cotangent is the ratio of the adjacent side to the opposite side.

$\cot \theta = \frac{\text{Adjacent}}{\text{Opposite}}$

where $0 < \theta < 90°$.

This ratio compares the adjacent side to the opposite side, which is the reverse of the tangent ratio.

When are these ratios undefined?

An important consideration is that these ratios become undefined when their denominators equal zero. Since division by zero is not possible in mathematics, we must identify when this occurs.

We'll explore these boundary cases in more detail later in these notes.

Exact values from special triangles

Just as we can find exact values for sine, cosine, and tangent at special angles, we can also determine exact values for secant, cosecant, and cotangent at the angles 30°, 45°, and 60°. These values come from two special right-angled triangles.

The 45-45-90 triangle

Consider an isosceles right-angled triangle where the two legs each have length 1 unit. This triangle has two 45° angles and one 90° angle.

Using Pythagoras' theorem to find the hypotenuse:

$c^2 = 1^2 + 1^2 = 2$

$c = \sqrt{2}$

From this triangle, we can calculate the exact values:

$\sec 45° = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{\sqrt{2}}{1} = \sqrt{2}$

$\operatorname{cosec} 45° = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{\sqrt{2}}{1} = \sqrt{2}$

$\cot 45° = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{1}{1} = 1$

The 30-60-90 triangle

To find exact values for 30° and 60° angles, we start with an equilateral triangle where all sides measure 2 units and all angles are 60°.

When we draw an altitude from one vertex to the opposite side, it bisects both the angle and the base. This creates two congruent right-angled triangles, each with angles of 30°, 60°, and 90°.

The altitude divides the base into two segments of 1 unit each. Using Pythagoras' theorem, we can find that the altitude has length $\sqrt{3}$ units.

Focusing on one of these right-angled triangles, we have a 30°-60°-90° triangle with:

• Hypotenuse = 2
• Opposite side to 30° angle = 1
• Adjacent side to 30° angle (opposite to 60°) = $\sqrt{3}$

From this triangle, we can calculate:

For 30°:

$\sec 30° = \frac{2}{\sqrt{3}}$

$\operatorname{cosec} 30° = \frac{2}{1} = 2$

$\cot 30° = \frac{\sqrt{3}}{1} = \sqrt{3}$

For 60°:

$\sec 60° = \frac{2}{1} = 2$

$\operatorname{cosec} 60° = \frac{2}{\sqrt{3}}$

$\cot 60° = \frac{1}{\sqrt{3}}$

Summary table of exact values

Here are all the exact values organised in a table for easy reference:

Angle	sec	cosec	cot
$30°$	$\frac{2}{\sqrt{3}}$	$2$	$\sqrt{3}$
$45°$	$\sqrt{2}$	$\sqrt{2}$	$1$
$60°$	$2$	$\frac{2}{\sqrt{3}}$	$\frac{1}{\sqrt{3}}$

Boundary values

The boundary values of $\sec \theta$, $\operatorname{cosec} \theta$, and $\cot \theta$ at $\theta = 0°$ and $\theta = 90°$ are found by examining what happens to the side ratios in a right-angled triangle as $\theta$ approaches these angles.

As θ → 0°

As the angle $\theta$ approaches 0°, the opposite side approaches 0 and the hypotenuse approaches the length of the adjacent side.

Let's examine each ratio:

Secant: $\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} \to \frac{\text{Hypotenuse}}{\text{Hypotenuse}} = 1$

Cosecant: $\operatorname{cosec} \theta = \frac{\text{Hypotenuse}}{\text{Opposite}} \to \frac{\text{Hypotenuse}}{0}$

This is undefined because we cannot divide by zero.

Cotangent: $\cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} \to \frac{\text{Hypotenuse}}{0}$

This is also undefined.

As θ → 90°

As the angle $\theta$ approaches 90°, the adjacent side approaches 0 and the opposite side approaches the length of the hypotenuse.

Let's examine each ratio:

Secant: $\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} \to \frac{\text{Hypotenuse}}{0}$

This is undefined because we cannot divide by zero.

Cosecant: $\operatorname{cosec} \theta = \frac{\text{Hypotenuse}}{\text{Opposite}} \to \frac{\text{Hypotenuse}}{\text{Hypotenuse}} = 1$

Cotangent: $\cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} \to \frac{0}{\text{Hypotenuse}} = 0$

Summary table of boundary values

These boundary values, which don't describe physical triangles, are important for understanding the complete behaviour of these functions:

Angle	sec	cosec	cot
$0°$	$1$	Undefined	Undefined
$90°$	Undefined	$1$	$0$

Worked examples

Remember!