Define and relate secant, cosecant, and cotangent

Ontario Grade 11 Mathematics

Trigonometric Functions

MCR3U – D1.4: Define and Relate Secant, Cosecant, and Cotangent

You already know three trigonometric ratios from Grade 10: sine, cosine, and tangent. Each of those ratios has a reciprocal — a companion ratio formed by flipping it upside down. These three companions are called secant, cosecant, and cotangent. They appear regularly in trigonometric identities and in the analysis of periodic functions, so being comfortable with them is an important step in MCR3U. This lesson builds directly on your knowledge of

\sin\theta

,

\cos\theta

, and

\tan\theta

, so keep those definitions close as you work through the new material.

What you will learn

Define secant, cosecant, and cotangent in terms of the primary trigonometric ratios.
Explain why each reciprocal ratio is undefined for certain angles.
Evaluate secant, cosecant, and cotangent for given angles using a calculator or known values.
Identify which primary ratio each reciprocal ratio is paired with and explain the relationship.

Prerequisite Bridge: The Three Primary Ratios

Recall from Grade 10 that for a right triangle with an acute angle

\theta

, the three primary ratios are defined using the sides of the triangle relative to that angle. The side directly across from

\theta

is called the opposite side, the side next to

\theta

(that is not the hypotenuse) is the adjacent side, and the longest side is the hypotenuse.

Using those labels:

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

,

\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

, and

\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

. These three ratios apply to any angle

\theta

in a right triangle, as long as

\theta

is between CAD 0° and CAD 90° (exclusive).

A key arithmetic fact you will need constantly: the reciprocal of a fraction

\frac{a}{b}

is

\frac{b}{a}

, provided neither

a

nor

b

is zero. For example, the reciprocal of

\frac{3}{5}

is

\frac{5}{3}

. Division by zero is never defined, so a reciprocal ratio is undefined whenever its paired primary ratio equals zero.

Opposite, adjacent, and hypotenuse are defined relative to the angle $\theta$ , not a fixed side.
The three primary ratios are $\sin\theta$ , $\cos\theta$ , and $\tan\theta$ .
A reciprocal is formed by swapping the numerator and denominator.
A ratio is undefined when its denominator equals zero.

Defining the Three Reciprocal Ratios

Each primary ratio has exactly one reciprocal ratio paired with it. The reciprocal of sine is called cosecant, written

\csc\theta

. The reciprocal of cosine is called secant, written

\sec\theta

. The reciprocal of tangent is called cotangent, written

\cot\theta

. Notice the abbreviations: csc, sec, and cot.

Formally, the definitions are:

\csc\theta = \frac{1}{\sin\theta}

, which equals

\frac{\text{hypotenuse}}{\text{opposite}}

. Next,

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

, which equals

\frac{\text{hypotenuse}}{\text{adjacent}}

. Finally,

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

, which equals

\frac{\text{adjacent}}{\text{opposite}}

.

A useful memory tip: the reciprocal of each ratio starts with the same letter as its pair's co-function. Cosecant pairs with sine (both involve 'co' and 's'), and secant pairs with cosine. Just remember that csc goes with sin, and sec goes with cos — they look like they should be switched, but they are not.

Because these ratios are reciprocals, two important relationships always hold for any angle where both sides are defined:

\sin\theta \cdot \csc\theta = 1

and

\cos\theta \cdot \sec\theta = 1

and

\tan\theta \cdot \cot\theta = 1

. Multiplying any ratio by its reciprocal always gives 1, the same way

\frac{3}{5} \times \frac{5}{3} = 1

.

\csc\theta = \frac{1}{\sin\theta}, \sec\theta = \frac{1}{\cos\theta}, \cot\theta = \frac{1}{\tan\theta}

$\csc\theta = \frac{1}{\sin\theta}$ — reciprocal of sine.
$\sec\theta = \frac{1}{\cos\theta}$ — reciprocal of cosine.
$\cot\theta = \frac{1}{\tan\theta}$ — reciprocal of tangent.
Each pair multiplies to 1: for example, $\cos\theta \cdot \sec\theta = 1$ .
Secant pairs with cosine; cosecant pairs with sine — not the other way around.

When Are the Reciprocal Ratios Undefined?

A reciprocal ratio is undefined whenever its paired primary ratio equals zero, because you cannot divide by zero. Let's check each one carefully.

Cosecant (

\csc\theta

) is undefined when

\sin\theta = 0

. In the range CAD 0° to CAD 360°, sine equals zero at

\theta = 0°

and

\theta = 180°

. At those angles, the opposite side has length zero, so the ratio hypotenuse over opposite has no meaning.

Secant (

\sec\theta

) is undefined when

\cos\theta = 0

. Cosine equals zero at

\theta = 90°

and

\theta = 270°

. At those angles, the adjacent side has length zero.

Cotangent (

\cot\theta

) is undefined when

\tan\theta = 0

, which also occurs at

\theta = 0°

and

\theta = 180°

. Notice that tangent itself is undefined at CAD 90° and CAD 270° (because its denominator,

\cos\theta

, is zero there). Cotangent, being the reciprocal of tangent, is actually defined at CAD 90° and CAD 270° — you just evaluate

\cot 90° = \frac{\cos 90°}{\sin 90°} = \frac{0}{1} = 0

.

An alternative formula for cotangent that avoids referencing tangent directly is

\cot\theta = \frac{\cos\theta}{\sin\theta}

. This comes from writing

\frac{1}{\tan\theta} = \frac{1}{\,\frac{\sin\theta}{\cos\theta}\,} = \frac{\cos\theta}{\sin\theta}

. This form makes it easier to spot when cotangent is undefined (when

\sin\theta = 0

) and when it equals zero (when

\cos\theta = 0

).

\cot\theta = \frac{\cos\theta}{\sin\theta}

$\csc\theta$ is undefined when $\sin\theta = 0$ , i.e., at CAD 0° and CAD 180°.
$\sec\theta$ is undefined when $\cos\theta = 0$ , i.e., at CAD 90° and CAD 270°.
$\cot\theta$ is undefined when $\sin\theta = 0$ , i.e., at CAD 0° and CAD 180°.
An equivalent form: $\cot\theta = \frac{\cos\theta}{\sin\theta}$ .

Evaluating Reciprocal Ratios on a Calculator

Most scientific calculators do not have dedicated buttons for csc, sec, and cot. Instead, you use the primary ratio and then take its reciprocal. Make sure your calculator is set to degree mode before you start.

To find

\sec 40°

, first calculate

\cos 40°

, then press the reciprocal button (usually labelled

x^{-1}

or

1/x

). So the sequence is:

\cos 40° \approx 0.7660

, then

\sec 40° = \frac{1}{0.7660} \approx 1.3054

.

The same strategy works for cosecant and cotangent. To find

\csc 25°

: calculate

\sin 25° \approx 0.4226

, then

\csc 25° = \frac{1}{0.4226} \approx 2.3662

. To find

\cot 70°

: calculate

\tan 70° \approx 2.7475

, then

\cot 70° = \frac{1}{2.7475} \approx 0.3640

.

Notice that when the primary ratio is less than 1 (and positive), its reciprocal is greater than 1. When the primary ratio is exactly 1, the reciprocal is also 1. The reciprocal ratio can never be between

-1

and

1

(exclusive) for secant and cosecant, because sine and cosine themselves are never greater than 1 in absolute value. Cotangent, however, can take any real value because tangent can.

\sec\theta = \frac{1}{\cos\theta}, \csc\theta = \frac{1}{\sin\theta}

Set the calculator to degree mode before evaluating any trig ratio.
Find csc, sec, or cot by evaluating the primary ratio first, then pressing the reciprocal button.
$|\sec\theta| \geq 1$ and $|\csc\theta| \geq 1$ for all defined angles.
$\cot\theta$ can equal any real number.

Connecting All Six Ratios: A Unified Picture

You now have six trigonometric ratios in total:

\sin\theta

,

\cos\theta

,

\tan\theta

,

\csc\theta

,

\sec\theta

, and

\cot\theta

. They are all connected, and knowing one ratio for an angle (along with the quadrant) is enough to find all six.

For example, if you know that

\sin\theta = \frac{3}{5}

and the angle is in the first quadrant, you can use the Pythagorean relationship

\sin^2\theta + \cos^2\theta = 1

to find

\cos\theta = \frac{4}{5}

. Then

\tan\theta = \frac{3/5}{4/5} = \frac{3}{4}

. The three reciprocal ratios follow immediately:

\csc\theta = \frac{5}{3}

,

\sec\theta = \frac{5}{4}

,

\cot\theta = \frac{4}{3}

.

This shows that the six ratios always come in three reciprocal pairs:

(\sin, \csc)

,

(\cos, \sec)

, and

(\tan, \cot)

. Understanding these pairs means you only ever need to remember three definitions — the reciprocal definitions give you the other three for free.

\sin^2\theta + \cos^2\theta = 1

All six ratios are determined once you know one ratio and the quadrant.
The three reciprocal pairs are: $(\sin, \csc)$ , $(\cos, \sec)$ , $(\tan, \cot)$ .
Each pair multiplies to 1 for any angle where both are defined.

The Six Trigonometric Ratios at a Glance

Ratio	Symbol	Definition (fraction)	Side Relationship	Paired With
Sine	$\sin\theta$	$\frac{1}{\csc\theta}$	opp / hyp	Cosecant
Cosine	$\cos\theta$	$\frac{1}{\sec\theta}$	adj / hyp	Secant
Tangent	$\tan\theta$	$\frac{1}{\cot\theta}$	opp / adj	Cotangent
Cosecant	$\csc\theta$	$\frac{1}{\sin\theta}$	hyp / opp	Sine
Secant	$\sec\theta$	$\frac{1}{\cos\theta}$	hyp / adj	Cosine
Cotangent	$\cot\theta$	$\frac{1}{\tan\theta}$	adj / opp	Tangent

Worked example

Evaluating All Six Ratios from a Right Triangle

A right triangle has legs of length 5 and 12, and a hypotenuse of length 13. The angle

\theta

is opposite the side of length 5. Find all six trigonometric ratios for

\theta

.

Identify the sides relative to angle theta
The side opposite $\theta$ has length 5. The side adjacent to $\theta$ has length 12. The hypotenuse is 13. Confirm it is a right triangle: $5^2 + 12^2 = 25 + 144 = 169 = 13^2$ . ✓
$5^2 + 12^2 = 13^2$
Write the three primary ratios
Using the definitions opposite/hypotenuse, adjacent/hypotenuse, and opposite/adjacent:
$\sin\theta = \frac{5}{13}, \cos\theta = \frac{12}{13}, \tan\theta = \frac{5}{12}$
Write the three reciprocal ratios
Flip each primary ratio to get its reciprocal. Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent:
$\csc\theta = \frac{13}{5}, \sec\theta = \frac{13}{12}, \cot\theta = \frac{12}{5}$
Verify one reciprocal pair multiplies to 1
Check that $\sin\theta \cdot \csc\theta = 1$ to confirm the reciprocal relationship:
$\frac{5}{13} × \frac{13}{5} = 1 \checkmark$

Answer:

\sin\theta = \frac{5}{13}

,

\cos\theta = \frac{12}{13}

,

\tan\theta = \frac{5}{12}

,

\csc\theta = \frac{13}{5}

,

\sec\theta = \frac{13}{12}

,

\cot\theta = \frac{12}{5}

Check: Each reciprocal pair multiplies to 1:

\frac{5}{13} \cdot \frac{13}{5} = 1

,

\frac{12}{13} \cdot \frac{13}{12} = 1

,

\frac{5}{12} \cdot \frac{12}{5} = 1

. All six values are confirmed correct.

Worked example

Using a Known Sine Value to Find the Remaining Five Ratios

Angle

\alpha

is in the first quadrant and

\sin\alpha = \frac{7}{25}

. Find

\cos\alpha

,

\tan\alpha

,

\csc\alpha

,

\sec\alpha

, and

\cot\alpha

. Then evaluate

\csc\alpha

as a decimal rounded to four decimal places.

Find cos alpha using the Pythagorean identity
Start with $\sin^2\alpha + \cos^2\alpha = 1$ . Substitute $\sin\alpha = \frac{7}{25}$ and solve for $\cos\alpha$ . Since $\alpha$ is in the first quadrant, cosine is positive.
$\cos\alpha = \sqrt{1 - (\frac{7}{25})^2} = \sqrt{1 - \frac{49}{625}} = \sqrt{\frac{576}{625}} = \frac{24}{25}$
Find tan alpha
Tangent equals sine divided by cosine:
$\tan\alpha = \frac{\sin\alpha}{\cos\alpha} = \frac{7/25}{24/25} = \frac{7}{24}$
Find the three reciprocal ratios
Flip each primary ratio. Cosecant is the reciprocal of $\frac{7}{25}$ , secant is the reciprocal of $\frac{24}{25}$ , and cotangent is the reciprocal of $\frac{7}{24}$ :
$\csc\alpha = \frac{25}{7}, \sec\alpha = \frac{25}{24}, \cot\alpha = \frac{24}{7}$
Convert csc alpha to a decimal
Divide 25 by 7 to get the decimal value of $\csc\alpha$ :
$\csc\alpha = \frac{25}{7} \approx 3.5714$
Verify using the reciprocal relationship
Check that $\sin\alpha \cdot \csc\alpha = 1$ :
$\frac{7}{25} × \frac{25}{7} = 1 \checkmark$

Answer:

\cos\alpha = \frac{24}{25}

,

\tan\alpha = \frac{7}{24}

,

\csc\alpha = \frac{25}{7} \approx 3.5714

,

\sec\alpha = \frac{25}{24}

,

\cot\alpha = \frac{24}{7}

Check: Verify the Pythagorean identity:

\left(\frac{7}{25}\right)^2 + \left(\frac{24}{25}\right)^2 = \frac{49}{625} + \frac{576}{625} = \frac{625}{625} = 1

. ✓ All five values are confirmed correct.

Common mistakes and how to avoid them

Thinking secant is the reciprocal of sine, and cosecant is the reciprocal of cosine — reversing the pairs.

Correction: Secant (

\sec

) is paired with cosine (

\cos

), and cosecant (

\csc

) is paired with sine (

\sin

). Remember: csc goes with sin.

Stating that

\cot 90°

is undefined because

\tan 90°

is undefined.

Correction:

\cot\theta = \frac{\cos\theta}{\sin\theta}

. At CAD 90°, this equals

\frac{0}{1} = 0

, so

\cot 90° = 0

. It is defined.

Forgetting to set the calculator to degree mode and getting an incorrect decimal answer.

Correction: Always confirm degree mode before evaluating. If the answer looks unreasonable (for example, a very large or very small number for a common angle), check the mode setting first.

Believing

\sec\theta

can equal

0.8

for some angle.

Correction: Because

|\cos\theta| \leq 1

, its reciprocal satisfies

|\sec\theta| \geq 1

. A value of

0.8

is between

-1

and

1

, so it is impossible for secant. The same applies to cosecant.

Writing

\csc\theta = \sin^{-1}\theta

, confusing reciprocal notation with inverse function notation.

Correction:

\sin^{-1}\theta

means the inverse sine function (used to find angles), while

\csc\theta = (\sin\theta)^{-1} = \frac{1}{\sin\theta}

is a ratio. These are completely different.

Lesson summary

Secant, cosecant, and cotangent are the reciprocals of cosine, sine, and tangent respectively.
Their symbols and definitions are: $\csc\theta = \frac{1}{\sin\theta}$ , $\sec\theta = \frac{1}{\cos\theta}$ , $\cot\theta = \frac{1}{\tan\theta}$ .
Each reciprocal pair multiplies to 1: for example, $\tan\theta \cdot \cot\theta = 1$ wherever both are defined.
A reciprocal ratio is undefined whenever its paired primary ratio equals zero — never divide by zero.
To evaluate on a calculator: find the primary ratio first, then press the reciprocal key. Always use degree mode.
The values of $|\sec\theta|$ and $|\csc\theta|$ are always greater than or equal to 1, while $\cot\theta$ can be any real number.

Check your understanding

Question 1

Which of the following correctly defines

\sec\theta

?

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

Show answer and explanation

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

Secant is the reciprocal of cosine, so

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

. Option A is cosecant, option C is tangent, and option D is cotangent.

Question 2

For which angle is

\csc\theta

undefined in the range CAD 0° to CAD 360°?

$\theta = 90°$
$\theta = 270°$
$\theta = 180°$
$\theta = 45°$

Show answer and explanation

\theta = 180°

\csc\theta = \frac{1}{\sin\theta}

, so it is undefined when

\sin\theta = 0

. In the range CAD 0° to CAD 360°,

\sin 180° = 0

, making

\csc 180°

undefined. At CAD 90° and CAD 270°, sine is not zero.

Question 3

A right triangle has

\cos\theta = \frac{8}{17}

. What is

\sec\theta

?

$\frac{8}{17}$
$\frac{17}{15}$
$\frac{15}{17}$
$\frac{17}{8}$

Show answer and explanation

\frac{17}{8}

Secant is the reciprocal of cosine. Flipping

\frac{8}{17}

gives

\sec\theta = \frac{17}{8}

. The other options involve incorrect side pairings or no reciprocal at all.

Question 4

Which statement about the value of

\sec\theta

is always true for any defined angle?

$\sec\theta$ is always between $-1$ and $1$ .
$|\sec\theta| \geq 1$ for all defined angles.
$\sec\theta$ is always positive.
$\sec\theta$ equals $\sin\theta$ when $\theta = 45°$ .

Show answer and explanation

|\sec\theta| \geq 1

for all defined angles.

Because

|\cos\theta| \leq 1

, its reciprocal satisfies

|\sec\theta| \geq 1

. Secant can be negative (e.g., in the second or third quadrant), so option C is false. At CAD 45°,

\sec 45° = \frac{1}{\cos 45°} = \sqrt{2} \neq \sin 45°

, so option D is also false.

Key terms

Reciprocal: The result of swapping the numerator and denominator of a fraction. The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$ , provided neither is zero.
Secant ( $\sec\theta$ ): The reciprocal of cosine: $\sec\theta = \frac{1}{\cos\theta}$ . In a right triangle it equals hypotenuse divided by adjacent.
Cosecant ( $\csc\theta$ ): The reciprocal of sine: $\csc\theta = \frac{1}{\sin\theta}$ . In a right triangle it equals hypotenuse divided by opposite.
Cotangent ( $\cot\theta$ ): The reciprocal of tangent: $\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$ . In a right triangle it equals adjacent divided by opposite.
Undefined ratio: A trigonometric ratio is undefined when its denominator equals zero, because division by zero has no meaning.
Degree mode: A calculator setting that interprets angles as degrees rather than any other unit. Always confirm this setting before evaluating trigonometric ratios in MCR3U.
Pythagorean identity: The identity $\sin^2\theta + \cos^2\theta = 1$ , which holds for every angle $\theta$ . It comes from the Pythagorean theorem applied to a unit right triangle.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation D1.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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