SSC CGL Maths · Trigonometric Ratios and Identities

Degree, Radian and Standard Values

The six ratios of an angle, their exact values at 0°, 30°, 45°, 60° and 90°, and how to turn degrees into radians and back.

All Trigonometric Ratios and Identities notes

Why this matters

34 questions in the bank, 32 of them EASY. Nine ask you to move between degrees and radians, and eight of those nine are from 2025, so the conversion is a recent favourite. The rest are a line of arithmetic once the table is in your head: the cheapest marks in the chapter, provided the table comes out in seconds.

Concept 1 of 4: The six trigonometric ratios

Pick one acute angle of a right triangle. Every ratio is a fraction of two of its sides, so there are six of them, and the last three are just the first three turned upside down.

Definition

Name the sides from the angle θ you are looking at:

The hypotenuse is the longest side, opposite the right angle.
The opposite side faces θ; the adjacent side touches θ and is not the hypotenuse.
$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$ , $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ , $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$ .
The reciprocals: $\text{cosec}\,\theta = \frac{1}{\sin\theta}$ , $\sec\theta = \frac{1}{\cos\theta}$ , $\cot\theta = \frac{1}{\tan\theta}$ .
Also $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and $\cot\theta = \frac{\cos\theta}{\sin\theta}$ .

The six ratios

\sin\theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan\theta = \frac{\text{opp}}{\text{adj}} = \frac{\sin\theta}{\cos\theta} \qquad \text{cosec}\,\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}

Worked example

In a right triangle, the side opposite θ is 8, the side adjacent to θ is 6 and the hypotenuse is 10. Write all six ratios of θ.

Practice this conceptself-check · 4 quick reps

Pairing cosec with cos

The names mislead:

\text{cosec}\,\theta

\frac{1}{\sin\theta}

and

\sec\theta

\frac{1}{\cos\theta}

. Each 'co' ratio is the reciprocal of the ratio without 'co'.

Turning tan the wrong way up

\tan\theta

is opposite over adjacent. Adjacent over opposite is

\cot\theta

, and it is always one of the printed wrong options.

Concept 2 of 4: Values of the six ratios at 0°, 30°, 45°, 60° and 90°

Five angles carry almost every SSC value question. The sine row climbs from 0 to 1, the cosine row is the same numbers read backwards, and the other four rows follow from those two.

Definition

A quick way to rebuild the sine row:

Write $\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}$ for 0°, 30°, 45°, 60°, 90°. That is $0, \frac{1}{2}, \frac{1}{\sqrt{2}}, \frac{\sqrt{3}}{2}, 1$ .
Cosine is the sine row in reverse.
tan is sine divided by cosine; cosec, sec and cot are the reciprocals.
From 0° to 90° sine rises and cosine falls. They are equal only at 45°, so $\sin\theta < \cos\theta$ below 45° and $\sin\theta > \cos\theta$ above it.
$\sin^2 60^\circ$ means $(\sin 60^\circ)^2 = \frac{3}{4}$ : square the value, never the angle.

Angle	sin	cos	tan
0°	$0$	$1$	$0$
30°	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
45°	$\frac{1}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}$	$1$
60°	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90°	$1$	$0$	Not defined tan 90° has no value, because cos 90° = 0 sits in the denominator.

cosec, sec and cot are the same entries flipped:

\text{cosec}\,30^\circ = 2

\sec 60^\circ = 2

\cot 30^\circ = \sqrt{3}

. A 0 flips to not defined, so cosec 0°, sec 90° and cot 0° have no value, and cot 90° = 0. Squares are friendlier:

\sin^2

runs

0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1

and

\tan^2

runs

0, \frac{1}{3}, 1, 3

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 21 Sep 2025, Shift 3 · Q69Easy

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

Find the value of

\sin^2 90^\circ + \cos^2 60^\circ

Swapping sin 60° and cos 60°

\sin 60^\circ = \frac{\sqrt{3}}{2}

and

\cos 60^\circ = \frac{1}{2}

. The bigger angle has the bigger sine, which is a fast way to check you have not swapped them.

Treating tan 90° as a number

\tan 90^\circ

and

\sec 90^\circ

are not defined, and neither are

\cot 0^\circ

and

\text{cosec}\,0^\circ

. An expression containing one of them has no value at that angle.

Drill 16 more on values of the six ratios at 0°, 30°, 45°, 60° and 90°

Concept 3 of 4: Reading the angle from a value

The table also works backwards: a value such as

\frac{1}{2}

for sine names the angle 30°. Once each bracket is a known angle, a pair of equations is just two lines of arithmetic.

Definition

Match the value to the table in the ratio that is given: $\sec\theta = 2$ means $\cos\theta = \frac{1}{2}$ , so $\theta = 60^\circ$ .
A bracket is one angle: $\cos(x + 10^\circ) = \frac{1}{2}$ gives $x + 10^\circ = 60^\circ$ .
Two equations in $A + B$ and $A - B$ : read both angles, then add and halve for A, subtract and halve for B.
The table answer is the acute one. SSC states 'acute' or a range such as $0^\circ < \theta < 90^\circ$ for exactly this reason.

Worked example

\tan(A + B) = \sqrt{3}

and

\tan(A - B) = \frac{1}{\sqrt{3}}

, where

A + B

is acute and

A > B

, find A and B.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 19 Sep 2025, Shift 3 · Q64Easy

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

\cot A = \sqrt{3}

, what is the value of

(1 + \sin A)(1 + \cos A)

Reading the angle off the wrong row

\sec\theta = 2

is 60°, not 30°: sec 30° is

\frac{2}{\sqrt{3}}

. Change a reciprocal ratio back to sin, cos or tan before you look the value up.

Taking the angle as the answer

When the question asks for an expression such as

\sin A + \cos 2A

, the angle is only the middle step. An option that prints the angle itself is a trap: put the values back in before you choose.

Drill 7 more on reading the angle from a value

Concept 4 of 4: Converting degrees and radians

A half turn is 180° and also π radians, so the two units differ by one fixed factor. Multiply by

\frac{180}{\pi}

to reach degrees and by

\frac{\pi}{180}

to reach radians.

Definition

$\pi \text{ rad} = 180^\circ$ , so $1 \text{ rad} = \frac{180^\circ}{\pi} \approx 57.3^\circ$ .
Radians to degrees: multiply by $\frac{180}{\pi}$ . With π in the angle, just put 180° in place of π: $\frac{3\pi}{4} = \frac{3 \times 180^\circ}{4} = 135^\circ$ .
Degrees to radians: multiply by $\frac{\pi}{180}$ and cancel: $45^\circ = \frac{\pi}{4}$ .
A plain number of radians (0.5, 1.5, 2.5) needs $\pi \approx 3.14159$ : $1.5 \text{ rad} \approx 85.94^\circ$ , not 90°.
A clock's minute hand turns 360° in 60 minutes, so 6° a minute. The hour hand turns 0.5° a minute.

Degree and radian conversion

\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180^\circ}{\pi} \qquad \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180^\circ}

Worked example

Convert

\frac{7\pi}{6}

radians to degrees and

315^\circ

to radians.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 2 · Q71Easy

Example 4 · Trigonometric Ratios and Identities · Degree, Radian and Standard Values

Convert 0.5 radians to degrees.

Multiplying by the wrong factor

To reach degrees multiply by

\frac{180}{\pi}

; to reach radians multiply by

\frac{\pi}{180}

. A quick check: degrees should come out as the bigger number, since 1 radian is about 57°.

Rounding 1 radian to 60°

1 radian is about 57.3°, so 1.5 rad is about 85.9° and 2.5 rad about 143.2°. Using 60° per radian lands on a printed wrong option such as 90° or 150°.

Drill 8 more on converting degrees and radians

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)

The six trigonometric ratios
The six ratios
$\sin\theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan\theta = \frac{\text{opp}}{\text{adj}} = \frac{\sin\theta}{\cos\theta} \qquad \text{cosec}\,\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}$
Converting degrees and radians
Degree and radian conversion
$\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180^\circ}{\pi} \qquad \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180^\circ}$

Reference tables (1)

Values of the six ratios at 0°, 30°, 45°, 60° and 90°5 rows

Angle	sin	cos	tan
0°	$0$	$1$	$0$
30°	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
45°	$\frac{1}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}$	$1$
60°	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90°	$1$	$0$	Not defined tan 90° has no value, because cos 90° = 0 sits in the denominator.

cosec, sec and cot are the same entries flipped:

\text{cosec}\,30^\circ = 2

\sec 60^\circ = 2

\cot 30^\circ = \sqrt{3}

. A 0 flips to not defined, so cosec 0°, sec 90° and cot 0° have no value, and cot 90° = 0. Squares are friendlier:

\sin^2

runs

0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1

and

\tan^2

runs

0, \frac{1}{3}, 1, 3

Watch out for (8)

Pairing cosec with cos
→ The six trigonometric ratios
Turning tan the wrong way up
→ The six trigonometric ratios
Swapping sin 60° and cos 60°
→ Values of the six ratios at 0°, 30°, 45°, 60° and 90°
Treating tan 90° as a number
→ Values of the six ratios at 0°, 30°, 45°, 60° and 90°
Reading the angle off the wrong row
→ Reading the angle from a value
Taking the angle as the answer
→ Reading the angle from a value
Multiplying by the wrong factor
→ Converting degrees and radians
Rounding 1 radian to 60°
→ Converting degrees and radians

Test yourself on Trigonometric Ratios and Identities

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Trigonometric Ratios and Identities chapter test Drill all 34 questions