MathsCBSE Class 11 · Trigonometric Functions

Radian Measure, Arc Length and Trigonometric Functions of Any Angle (Class 11)

Updated 17 October 2026 · Fact-checked

An angle of 180° equals π radians, and an arc of length l on a circle of radius r subtends θ = l/r radians. For any angle, the signs of the ratios follow the quadrant (All, Sine, Tan, Cos: ASTC). Allied angles reduce to acute ones, for example sin(π − x) = sin x and cos(π + x) = −cos x. The functions sin and cos have period 2π and range [−1, 1]; tan has period π.

Understand Radian Measure and Trigonometric Functions of Any Angle

Radian measure. One radian is the angle subtended at the centre by an arc equal to the radius. Hence
$ $π rad = 18 0^{\circ}, radians = degrees \times \frac{π}{180}, θ = \frac{l}{r} .$ $
For $4 0^{\circ} 2 0^{'}$ change minutes to degrees first: $40 \frac{1}{3}^{\circ} = \frac{121}{3}^{\circ}$ , which is $\frac{121 π}{540}$ rad. A sector with radius $r$ and angle $θ$ has arc $l = r θ$ and area $\frac{1}{2} r^{2} θ = \frac{1}{2} r l$ ; its perimeter is $2 r + l$ . The angle $θ$ must be in radians in these formulas.

Functions on the unit circle. A point on the unit circle at angle $x$ has coordinates $(cos x, sin x)$ ; $tan x = \frac{s i n x}{c o s x}$ (undefined where $cos x = 0$ ), and $sec, cosec, cot$ are the reciprocals of $cos, sin, tan$ .

Signs (ASTC). Quadrant I: all positive. II: only $sin$ (and $cosec$ ). III: only $tan$ (and $cot$ ). IV: only $cos$ (and $sec$ ).

Allied angles. Reduce to $0$ to $\frac{π}{2}$ using the quadrant sign:

$sin (π - x) = sin x$ , $cos (π - x) = - cos x$ ; $sin (π + x) = - sin x$ , $cos (π + x) = - cos x$ ;
$sin (2 π - x) = - sin x$ , $cos (2 π - x) = cos x$ ; $sin (- x) = - sin x$ , $cos (- x) = cos x$ ;
$sin (\frac{π}{2} - x) = cos x$ , $cos (\frac{π}{2} - x) = sin x$ .

Subtract multiples of $36 0^{\circ}$ for large angles: $sin 60 0^{\circ} = sin 24 0^{\circ} = - \frac{3}{2}$ .

Finding other ratios. Given one ratio and the quadrant, draw the right triangle for the reference angle, find the third side by Pythagoras, then attach signs. For $tan x = - \frac{5}{12}$ in quadrant II: hypotenuse $13$ , so $sin x = \frac{5}{13}$ and $cos x = - \frac{12}{13}$ .

Domain, range, period. $sin, cos$ : domain $R$ , range $[- 1, 1]$ , period $2 π$ . $tan$ : domain $R - {(2 k + 1) \frac{π}{2}}$ , range $R$ , period $π$ . The period of $sin k x$ or $cos k x$ is $\frac{2 π}{∣ k ∣}$ ; the period of a sum is the least common multiple of the periods when it exists. Simplify before judging: $sin^{4} x - cos^{4} x = - cos 2 x$ , which has period $π$ and range $[- 1, 1]$ .

Radian Measure and Trigonometric Functions of Any Angle: formulas to remember

Degrees and radians: $180° = π r a d ian s$; radians = degrees × π/180.
Arc length and sector: $l = r θ; a r e a = (1/2) r^{2} θ$; θ in radians.
Pythagorean identity: $s i n^{2} x + co s^{2} x = 1; 1 + t a n^{2} x = se c^{2} x$; Also 1 + cot² x = cosec² x.
Allied angles: $s in (π - x) = s in x; cos (π + x) = - cos x$; Sign from the quadrant.
Period of sin kx: $2 π /∣ k ∣$; tan kx: π/|k|.

How to solve Radian Measure and Trigonometric Functions of Any Angle questions

Use these steps for conversions, arcs and ratios of any angle.

1Convert minutes and seconds to degrees, then to radians if the formula needs it.
2For arcs and sectors use $l = r θ$ with $θ$ in radians; state units.
3For a large or negative angle subtract multiples of $36 0^{\circ}$ (or $2 π$ ), then use allied angles.
4Find the quadrant and write the sign of the required ratio (ASTC).
5Given one ratio, draw the triangle, find the third side and attach the signs.
6For periods, find the period of each term and take the LCM; simplify the expression first when possible.

Radian Measure and Trigonometric Functions of Any Angle in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Arc length on a Ferris wheel

Suppose a Ferris wheel has radius 20 m and a boarding platform moves a cabin through $5 4^{\circ}$ of the circle.

How far does the cabin travel along the circle?

Try it first, then show the solutionSolution

$θ = 54 \times \frac{π}{180} = \frac{3 π}{10}$ rad.
$l = r θ = 20 \times \frac{3 π}{10} = 6 π \approx 18.85$ m.

Answer: About 18.85 m.

Memory hook: Arc = radius × angle in radians.

Ratios from a slope

Suppose a ramp makes an angle $x$ with the ground such that $tan x = \frac{5}{12}$ .

How high does a 26 m long ramp rise?

Try it first, then show the solutionSolution

Hypotenuse for sides $5, 12$ : $25 + 144 = 13$ , so $sin x = \frac{5}{13}$ .
Height $= 26 sin x = 26 \times \frac{5}{13} = 10$ m.

Answer: 10 m

Memory hook: Find the missing side first; then read the ratio.

Tricks to remember Radian Measure and Trigonometric Functions of Any Angle

Short memory aids. Say each one aloud once, then use it on a question the same day.

ASTC
All, Sine, Tan, Cos: which ratios are positive in quadrants I to IV.
Use it when: Signs.
Radians for arcs
The formula l = r θ only works with θ in radians.
Use it when: Arc and sector questions.
Reduce then sign
Strip multiples of 360°, reduce to an acute angle, then attach the sign.
Use it when: Large angles.
Period of sin kx
2π divided by the absolute value of k.
Use it when: Period questions.

Radian Measure and Trigonometric Functions of Any Angle: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Conversion · 2 marks · Very short answerBoard level

Convert $4 0^{\circ} 2 0^{'}$ to radians and $\frac{5 π}{12}$ radians to degrees.

Try it first, then show the solutionSolution

$4 0^{\circ} 2 0^{'} = 40 \frac{1}{3}^{\circ} = \frac{121}{3}^{\circ}$ ; in radians $\frac{121}{3} \times \frac{π}{180} = \frac{121 π}{540}$ . (1 mark)
$\frac{5 π}{12} \times \frac{180}{π} = 7 5^{\circ}$ . (1 mark)

Answer: $\frac{121 π}{540}$ rad; $7 5^{\circ}$

Example 2: Sector · 3 marks · Short answerBoard level

A sector of a circle of radius 6 cm has perimeter 30 cm. Find the angle at the centre in radians and the area of the sector.

Try it first, then show the solutionSolution

Perimeter $= 2 r + l$ , so $l = 30 - 12 = 18$ cm. (1 mark)
$θ = \frac{l}{r} = \frac{18}{6} = 3$ radians. (1 mark)
Area $= \frac{1}{2} r l = \frac{1}{2} \times 6 \times 18 = 54$ cm². (1 mark)

Answer: 3 rad; 54 cm²

Example 3: Value of an expression · 3 marks · Short answerHarder board level

Find the value of $sin 60 0^{\circ} cos 33 0^{\circ} + cos 12 0^{\circ} sin 15 0^{\circ}$ .

Try it first, then show the solutionSolution

$sin 60 0^{\circ} = sin 24 0^{\circ} = - \frac{3}{2}$ , $cos 33 0^{\circ} = \frac{3}{2}$ . (1 mark)
$cos 12 0^{\circ} = - \frac{1}{2}$ , $sin 15 0^{\circ} = \frac{1}{2}$ . (1 mark)
Value $= - \frac{3}{4} - \frac{1}{4} = - 1$ . (1 mark)

Answer: $- 1$

Example 4: All ratios from one ratio · 3 marks · Short answerHarder board level

If $sec x = \frac{13}{5}$ and $x$ lies in the fourth quadrant, find the other five trigonometric ratios.

Try it first, then show the solutionSolution

$cos x = \frac{5}{13}$ ; reference triangle with adjacent $5$ and hypotenuse $13$ has opposite side $169 - 25 = 12$ . (1 mark)
In quadrant IV only $cos$ and $sec$ are positive: $sin x = - \frac{12}{13}$ , $tan x = - \frac{12}{5}$ . (1 mark)
$cosec x = - \frac{13}{12}$ , $cot x = - \frac{5}{12}$ . (1 mark)

Answer: $cos = \frac{5}{13}$ , $sin = - \frac{12}{13}$ , $tan = - \frac{12}{5}$ , $cosec = - \frac{13}{12}$ , $cot = - \frac{5}{12}$

Example 5: Simplify, then find period and range · 5 marks · Long answerHOTS / competency

Show that $f (x) = sin^{4} x - cos^{4} x = - cos 2 x$ . Hence find the period and range of $f$ , and $f (\frac{π}{6})$ .

Try it first, then show the solutionSolution

$sin^{4} x - cos^{4} x = (sin^{2} x - cos^{2} x) (sin^{2} x + cos^{2} x)$ . (1 mark)
$= (sin^{2} x - cos^{2} x) \cdot 1 = - cos 2 x$ . (1 mark)
Period of $cos 2 x$ is $\frac{2 π}{2} = π$ . (1 mark)
Range of $- cos 2 x$ is $[- 1, 1]$ . (1 mark)
$f (\frac{π}{6}) = - cos \frac{π}{3} = - \frac{1}{2}$ . Check: $\frac{1}{16} - \frac{9}{16} = - \frac{1}{2}$ . (1 mark)

Answer: $f (x) = - cos 2 x$ ; period $π$ ; range $[- 1, 1]$ ; $f (\frac{π}{6}) = - \frac{1}{2}$

Where students lose marks in Radian Measure and Trigonometric Functions of Any Angle

The slips examiners see most often, and what to write instead.

Mistake: Using degrees in l = r θ
The formula needs radians.
Wrong working
$l = 20 \times 54 = 1080$ .
Right working
$l = 20 \times \frac{3 π}{10} = 6 π$ .
Do this: Convert the angle to radians first.
Usually costs: The answer.
Mistake: Ignoring the quadrant sign
Only the magnitude is found from the triangle.
Wrong working
$tan x = - \frac{5}{12}$ in quadrant II: $cos x = \frac{12}{13}$ .
Right working
$cos x = - \frac{12}{13}$ .
Do this: Apply ASTC before writing the answer.
Usually costs: 1 to 2 marks.
Mistake: Wrong allied-angle sign
The sign of the original function in the new quadrant is forgotten.
Wrong working
$cos (π - x) = cos x$ .
Right working
$cos (π - x) = - cos x$ .
Do this: Name the quadrant and its sign.
Usually costs: The answer.
Mistake: Taking the sum of the periods
The period of a sum is the LCM, not the sum.
Wrong working
Period of $sin 3 x + cos 2 x$ $= \frac{2 π}{3} + π$ .
Right working
LCM $= 2 π$ .
Do this: List multiples of each period.
Usually costs: The answer.

Exam tips

Convert minutes to degrees (divide by 60) before converting to radians.
Write the quadrant and ASTC sign before finding the value.
Draw the reference triangle for every "find the other ratios" question.
Show the reduction of large angles step by step (for example $60 0^{\circ} - 36 0^{\circ} = 24 0^{\circ}$ ).
Simplify an expression before asking for its period or range.

Practice questions on Radian Measure and Trigonometric Functions of Any Angle

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
$15 0^{\circ}$ in radians is:
(a) $\frac{5 π}{6}$
(b) $\frac{2 π}{3}$
(c) $\frac{3 π}{4}$
(d) $\frac{5 π}{12}$
Hint
Multiply by $\frac{π}{180}$ .
Answer
(a). $\frac{150 π}{180} = \frac{5 π}{6}$ .
Q2 · 1 mark · MCQHarder board level
If $sin x = \frac{3}{5}$ and $x$ lies in the second quadrant, then $tan x$ is:
(a) $\frac{3}{4}$
(b) $- \frac{3}{4}$
(c) $\frac{4}{3}$
(d) $- \frac{4}{3}$
Hint
Find $cos x$ with its sign.
Answer
(b). $cos x = - \frac{4}{5}$ , so $tan x = \frac{3/5}{- 4/5} = - \frac{3}{4}$ .
Q3 · 3 marks · Short answerHarder board level
An arc of length 22 cm is cut from a circle of radius 14 cm. Find the angle subtended at the centre in radians and in degrees (take $π = \frac{22}{7}$ ).
Hint
Use $θ = \frac{l}{r}$ .
Answer
$θ = \frac{22}{14} = \frac{11}{7}$ rad. In degrees: $\frac{11}{7} \times \frac{180}{π} = \frac{11}{7} \times 180 \times \frac{7}{22} = 9 0^{\circ}$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): $sin (π - x) = sin x$ for every real $x$ .
Reason (R): The angles $x$ and $π - x$ correspond to points on the unit circle with the same $y$ -coordinate.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
The point for $π - x$ is the reflection of the point for $x$ in the $y$ -axis.
Answer
(a). Reflection in the $y$ -axis keeps the $y$ -coordinate, which is the sine.
Q5 · 4 marks · Short answerHOTS / competency
Find the period of $f (x) = sin 3 x + cos 2 x$ .
Hint
Find the period of each term, then the LCM.
Answer
$sin 3 x$ has period $\frac{2 π}{3}$ (also $\frac{4 π}{3}, 2 π, \dots$ ); $cos 2 x$ has period $π$ (also $2 π, \dots$ ). The least common value is $2 π$ . Check: $f (x + 2 π) = f (x)$ , while neither $\frac{2 π}{3}$ nor $π$ works for the sum.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.How many degrees is π radians?Show answer
180°.
2.What is l in terms of r and θ?Show answer
l = r θ (θ in radians).
3.In which quadrants is tan positive?Show answer
I and III.
4.What is sin(π + x)?Show answer
−sin x.
5.What is the period of tan x?Show answer
π.

Radian Measure and Trigonometric Functions of Any Angle: frequently asked questions

Why do we use radians?

Radians make arc-length and later calculus formulas simple, like l = r θ.

How do I remember the signs?

Use ASTC: All positive in quadrant I, Sine in II, Tan in III, Cos in IV.

What is the range of tan x?

All real numbers.

Are the graphs of the functions examined?

Yes, domain, range, period and the shapes of the graphs of sin, cos and tan belong to this chapter.