CDS Mathematics · Trigonometric Ratios and Identities

Degree, Radian & Standard Values

The raw material of the chapter: converting between degrees and radians, the ratios of 0°, 30°, 45°, 60° and 90°, and what values a ratio can and cannot take.

All Trigonometric Ratios and Identities notes CDS Mathematics strategy

Why this matters

Twenty PYQs and the cheapest page in the chapter — seven are EASY and only one is HARD. Every other page assumes these values are instant, so an hour here pays back across all 227 questions in the chapter.

Concept 1 of 3: Degree and radian measure, and arc length

A radian is the angle that cuts off an arc exactly one radius long. A full turn fits

2\pi

radii round the circle, so a full turn is

2\pi

radians as well as

360^\circ

. One radian is therefore a big angle — about

57.3^\circ

— which is the fact behind every 'compare

\sin 1^\circ

with

\sin 1^c

' question.

Definition

Conversion: $\pi$ radians $= 180^\circ$ , so multiply degrees by $\dfrac{\pi}{180}$ to get radians, and radians by $\dfrac{180}{\pi}$ to get degrees. A superscript $c$ (as in $1^c$ ) means radians.

$1^c \approx 57.3^\circ$ , and $1^\circ \approx 0.0175$ radian.
Arc length $s = r\theta$ , with $\theta$ in radians.
One revolution is $2\pi$ radians, so $N$ revolutions turn $2\pi N$ radians.

Degree–radian conversion and arc length

\pi \text{ rad} = 180^\circ, \qquad s = r\theta \;(\theta \text{ in radians})

Worked example

Two angles add up to

\dfrac{\pi}{2}

radian and differ by

30^\circ

. Find the larger angle in degrees.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (I) 2025 — Elementary Mathematics · Q27Easy

Example 1 · Trigonometric Ratios and Identities · Degree, Radian and Standard Values

The length of an arc of a circle of radius 4 cm is

\pi

cm. What is the magnitude of the angle subtended by the arc at the centre ?

Arc length needs the angle in radians

s = r\theta

is only true with

\theta

in radians. Putting

60

(degrees) into it gives

360

cm for a

6

cm circle — an arc longer than the whole circumference. Convert first.

One radian is not a small angle

Students read

1^c

as 'about one degree'. It is about

57.3^\circ

, so

\sin 1^c \approx 0.84

while

\sin 1^\circ \approx 0.017

. Every comparison between the two turns on this.

Drill 6 more on degree and radian measure, and arc length

Concept 2 of 3: Ratios of the standard angles

Five angles —

0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ

— are the only ones the paper expects you to know by value. Sine climbs through

\dfrac{\sqrt0}{2}, \dfrac{\sqrt1}{2}, \dfrac{\sqrt2}{2}, \dfrac{\sqrt3}{2}, \dfrac{\sqrt4}{2}

; cosine runs the same list backwards; everything else is a quotient or a reciprocal.

Definition

Learn the sine row as $\dfrac{\sqrt{n}}{2}$ for $n = 0, 1, 2, 3, 4$ . Then:

cosine is the sine row reversed;
$\tan = \dfrac{\sin}{\cos}$ , and $\cot$ , $\sec$ , $\operatorname{cosec}$ are reciprocals of $\tan$ , $\cos$ , $\sin$ .

A value that is 'not defined' comes from dividing by zero: $\tan 90^\circ$ , $\sec 90^\circ$ , $\cot 0^\circ$ and $\operatorname{cosec} 0^\circ$ .

Angle	sin	cos	tan
$0^\circ$	$0$	$1$	$0$
$30^\circ$	$\dfrac{1}{2}$	$\dfrac{\sqrt3}{2}$	$\dfrac{1}{\sqrt3}$
$45^\circ$	$\dfrac{1}{\sqrt2}$	$\dfrac{1}{\sqrt2}$	$1$
$60^\circ$	$\dfrac{\sqrt3}{2}$	$\dfrac{1}{2}$	$\sqrt3$
$90^\circ$	$1$	$0$	not defined $\tan 90^\circ$ and $\sec 90^\circ$ are not defined — they are not 0 and not 1.

Sine climbs as

\sqrt{n}/2

, cosine is the same list reversed, and tangent is their quotient.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (II) 2017 — Elementary Mathematics · Q50Moderate

Example 2 · Trigonometric Ratios and Identities · Degree, Radian and Standard Values

A = \frac{\sin 45^\circ - \sin 30^\circ}{\cos 45^\circ + \cos 60^\circ}

and

B = \frac{\sec 45^\circ - \tan 45^\circ}{\operatorname{cosec} 45^\circ + \cot 45^\circ}

, then which one of the following is correct ?

Rationalise before comparing

\dfrac{\sqrt2 - 1}{\sqrt2 + 1}

and

3 - 2\sqrt2

are the same number. When two expressions in a question look different, rationalise both before deciding they differ.

Drill 5 more on ratios of the standard angles

Concept 3 of 3: What values a ratio can take, and its sign

Sine and cosine are coordinates of a point on a circle of radius 1, so they can never leave

[-1, 1]

. Their reciprocals can therefore never enter

(-1, 1)

. Tangent is unrestricted. Most 'which of these is possible?' questions are settled by that one sentence before any algebra.

Definition

$-1 \le \sin\theta \le 1$ and $-1 \le \cos\theta \le 1$ .
$|\sec\theta| \ge 1$ and $|\operatorname{cosec}\theta| \ge 1$ .
$\tan\theta$ and $\cot\theta$ take every real value.
Signs: all ratios are positive in the first quadrant; sine (and cosec) in the second; tangent (and cot) in the third; cosine (and sec) in the fourth.
In the first quadrant sine and tangent increase with the angle and cosine decreases, so a larger cosine means a smaller angle. Below $45^\circ$ the cotangent exceeds the tangent; above $45^\circ$ the tangent exceeds the cotangent.

Ranges

|\sin\theta|, |\cos\theta| \le 1, \qquad |\sec\theta|, |\operatorname{cosec}\theta| \ge 1

Worked example

\sin\alpha + \sin\beta + \sin\gamma = 3

, find

\cos\alpha + \cos\beta + \cos\gamma

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (II) 2019 — Elementary Mathematics · Q60Moderate

Example 3 · Trigonometric Ratios and Identities · Degree, Radian and Standard Values

Consider the following for real numbers

\alpha, \beta, \gamma

and

\delta

: 1.

\sec \alpha = 1/4

\tan \beta = 20

\operatorname{cosec} \gamma = 1/2

\cos \delta = 2

How many of the above statements are not possible ?

A larger cosine means a smaller angle

0^\circ

90^\circ

the cosine falls as the angle rises. So

\cos\theta < \cos\phi

means

\theta > \phi

. Reading it the way sine behaves reverses the answer.

Drill 6 more on what values a ratio can take, and its sign

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Degree and radian measure, and arc length
Degree–radian conversion and arc length
$\pi \text{ rad} = 180^\circ, \qquad s = r\theta \;(\theta \text{ in radians})$
What values a ratio can take, and its sign
Ranges
$|\sin\theta|, |\cos\theta| \le 1, \qquad |\sec\theta|, |\operatorname{cosec}\theta| \ge 1$

Reference tables (1)

Ratios of the standard angles5 rows

Angle	sin	cos	tan
$0^\circ$	$0$	$1$	$0$
$30^\circ$	$\dfrac{1}{2}$	$\dfrac{\sqrt3}{2}$	$\dfrac{1}{\sqrt3}$
$45^\circ$	$\dfrac{1}{\sqrt2}$	$\dfrac{1}{\sqrt2}$	$1$
$60^\circ$	$\dfrac{\sqrt3}{2}$	$\dfrac{1}{2}$	$\sqrt3$
$90^\circ$	$1$	$0$	not defined $\tan 90^\circ$ and $\sec 90^\circ$ are not defined — they are not 0 and not 1.

Sine climbs as

\sqrt{n}/2

, cosine is the same list reversed, and tangent is their quotient.

Watch out for (4)

Arc length needs the angle in radians
→ Degree and radian measure, and arc length
One radian is not a small angle
→ Degree and radian measure, and arc length
Rationalise before comparing
→ Ratios of the standard angles
A larger cosine means a smaller angle
→ What values a ratio can take, and its sign

Test yourself on Trigonometric Ratios and Identities

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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