Trigonometric Functions

Q: What is the domain and range of sin x and cos x?

Both sin x and cos x have domain ℝ (all real numbers) and range [-1, 1]. They are the only two trigonometric functions defined for every real angle.

Q: Why is tan x not defined at π/2?

Because tan x = sin x / cos x and cos(π/2) = 0, division by zero makes tan(π/2) undefined. The graph shows a vertical asymptote at x = π/2 and at every x = (2n+1)π/2.

Q: What is the period of trigonometric functions?

sin x, cos x, csc x, and sec x have period 2π. tan x and cot x have period π. This means the graphs repeat exactly after these intervals.

Q: How do you convert degrees to radians?

Use the relation 180° = π radians. So degree to radian: multiply by π/180. Radian to degree: multiply by 180/π. For example 60° = 60 × π/180 = π/3 radians.

Q: What is 1 radian in degrees?

1 radian = 180/π ≈ 57.2958°. A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.

Q: What are the values of sin, cos, tan at standard angles?

Key values: sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1. cos values are the same in reverse order. tan at these angles: 0, 1/√3, 1, √3, undefined.

MathsTrigonometryFor JEE aspirants

Trigonometric functions extend the sine, cosine and tangent ratios from right triangles to angles of any size using the unit circle. The six functions are $sin x$ , $cos x$ , $tan x$ , $cot x$ , $sec x$ and $csc x$ . This page covers angle measurement (degree, grade, radian), the six trigonometric functions with their domain, range and graphs, signs in the four quadrants (ASTC rule), values at standard angles, allied-angle formulas and periodicity - the foundation on which all of JEE trigonometry rests.

Key Formulas - Quick Reference

Radian–degree: $180° = π$ rad; so $1° = π /180$ rad and $1$ rad $= 180/ π \approx 57.3°$
Arc length $ℓ = r θ$ ; sector area $= \frac{1}{2} r^{2} θ$ (with $θ$ in radians)
Pythagorean identities: $sin^{2} x + cos^{2} x = 1$ , $1 + tan^{2} x = sec^{2} x$ , $1 + cot^{2} x = csc^{2} x$
Reciprocals: $csc x = 1/ sin x$ , $sec x = 1/ cos x$ , $cot x = 1/ tan x = cos x / sin x$
Periods: $sin, cos, csc, sec$ have period $2 π$ ; $tan, cot$ have period $π$
Domain restrictions: $tan x, sec x$ undefined at $x = (2 n + 1) π /2$ ; $cot x, csc x$ undefined at $x = nπ$

1. Angle and its Measurement

An angle is a measure of rotation of a ray about its initial point. The starting position is the initial side, the final position after rotation is the terminal side, and the point of rotation is the vertex. Anticlockwise rotation gives a positive angle; clockwise rotation gives a negative angle.

1.1 Three Systems of Measurement

System	Unit	One right angle	Subdivisions
Sexagesimal (British)	degree (°)	$90°$	$1° = 6 0^{'}$ , $1^{'} = 6 0^{''}$
Centesimal (French)	grade ( $^{g}$ )	$10 0^{g}$	$1^{g} = 10 0^{'}$ , $1^{'} = 10 0^{''}$
Circular (Radian)	radian (rad)	$π /2$ rad	-

Minutes and seconds in the Sexagesimal and Centesimal systems are different - the symbols look similar but the sizes differ. When no symbol is written, an angle is assumed to be in radians. For example, "

θ = 15

" means

15

radians, not

15

degrees.

1.2 Radian - the Natural Unit

One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. Because the circumference of a unit circle is $2 π$ , one full revolution equals $2 π$ radians. In a circle of radius $r$ , an arc of length $ℓ$ subtends the angle

θ = \frac{ℓ}{r} ⟺ ℓ = r θ (θ in radians)

The relation between the three systems follows from a right angle:

\frac{π}{2} rad = 90° = 10 0^{g}

1.3 Arc Length and Sector Area

For a circular sector of radius

r

and central angle

θ

(radians):

arc length = r θ, sector area = \frac{1}{2} r^{2} θ

Solved Example 1

Convert

210°

to radians and find the length of the arc it cuts on a circle of radius

6

cm.

Solution:

$210° = 210 \times \frac{π}{180} = \frac{7 π}{6}$ rad.

Arc length $= r θ = 6 \cdot \frac{7 π}{6} = 7 π$ cm.

2. Trigonometric Ratios of an Acute Angle

Take a right triangle $A B C$ with the right angle at $B$ and angle $θ$ at $A$ . Label the sides base (adjacent to $θ$ ), perpendicular (opposite $θ$ ) and hypotenuse (opposite the right angle). The six trigonometric ratios are:

Figure: Right triangle used to define

sin θ

cos θ

tan θ

and reciprocals.

Six Ratios in a Right Triangle

$sin θ = \frac{perpendicular}{hypotenuse}$ , $csc θ = \frac{hypotenuse}{perpendicular}$
$cos θ = \frac{base}{hypotenuse}$ , $sec θ = \frac{hypotenuse}{base}$
$tan θ = \frac{perpendicular}{base}$ , $cot θ = \frac{base}{perpendicular}$

The three Pythagorean identities follow directly from $a^{2} + b^{2} = c^{2}$ :

sin^{2} θ + cos^{2} θ = 1, 1 + tan^{2} θ = sec^{2} θ, 1 + cot^{2} θ = csc^{2} θ

3. Extension to Any Real Angle - the Unit Circle

For angles beyond the acute range, we use the unit circle (radius $1$ , centred at origin). If a radius makes angle $x$ with the positive $x$ -axis and ends at point $P (a, b)$ , we define:

Figure: Unit circle definition - for a point

P (cos x, sin x)

on the unit circle.

cos x = a, sin x = b

Because

P

lies on the unit circle,

a^{2} + b^{2} = 1

, giving

cos^{2} x + sin^{2} x = 1

for every real

x

The other four functions inherit from these:

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

3.1 Quadrantal Angles

At the axes points $A (1, 0), B (0, 1), C (- 1, 0), D (0, - 1)$ the values are:

Angle	$0$	$π /2$	$π$	$3 π /2$	$2 π$
$sin x$	$0$	$1$	$0$	$- 1$	$0$
$cos x$	$1$	$0$	$- 1$	$0$	$1$

4. Signs of Trigonometric Functions - the ASTC Rule

The sign of each function depends on the signs of $a$ and $b$ , i.e. on the quadrant of $P$ . The mnemonic ASTC (read anticlockwise from quadrant I: All, Sine, Tan, Cos) records which functions are positive in each quadrant.

Figure: ASTC rule - sign of trigonometric ratios in each quadrant.

Quadrant	Positive functions	Negative functions
I	all six	none
II	$sin$ , $csc$	$cos$ , $sec$ , $tan$ , $cot$
III	$tan$ , $cot$	$sin$ , $csc$ , $cos$ , $sec$
IV	$cos$ , $sec$	$sin$ , $csc$ , $tan$ , $cot$

5. Values at Standard Angles

$θ$	$0$	$π /6$	$π /4$	$π /3$	$π /2$
$sin θ$	$0$	$1/2$	$1/ 2$	$3 /2$	$1$
$cos θ$	$1$	$3 /2$	$1/ 2$	$1/2$	$0$
$tan θ$	$0$	$1/ 3$	$1$	$3$	N.D.

N.D. = not defined. Cosec, sec and cot values are reciprocals of sin, cos and tan respectively.

6. Allied Angles

If $θ$ is any angle, then $- θ$ , $π /2 \pm θ$ , $π \pm θ$ , $3 π /2 \pm θ$ and $2 π \pm θ$ are called allied angles. Their trigonometric values are related to those of $θ$ by two simple rules:

Function name changes (co-function) when the angle involves $π /2$ or $3 π /2$ (odd multiples of $π /2$ ): $sin \leftrightarrow cos$ , $tan \leftrightarrow cot$ , $sec \leftrightarrow csc$ .
Function name is unchanged when the angle involves $π$ or $2 π$ (even multiples of $π /2$ ).
The sign is determined by the ASTC quadrant of the allied angle (treating $θ$ as acute).

6.1 Ratios of $- θ$

sin (- θ) = - sin θ, cos (- θ) = cos θ, tan (- θ) = - tan θ

$csc, sec, cot$ follow the same pattern as their reciprocals.

6.2 Ratios of $π /2 - θ$ (complementary angles)

sin (\frac{π}{2} - θ) = cos θ, cos (\frac{π}{2} - θ) = sin θ, tan (\frac{π}{2} - θ) = cot θ

6.3 Ratios of $π /2 + θ$

sin (\frac{π}{2} + θ) = cos θ, cos (\frac{π}{2} + θ) = - sin θ, tan (\frac{π}{2} + θ) = - cot θ

6.4 Ratios of $π \pm θ$

sin (π - θ) = sin θ, cos (π - θ) = - cos θ, tan (π - θ) = - tan θ

sin (π + θ) = - sin θ, cos (π + θ) = - cos θ, tan (π + θ) = tan θ

6.5 Ratios of $3 π /2 \pm θ$

sin (\frac{3 π}{2} - θ) = - cos θ, cos (\frac{3 π}{2} - θ) = - sin θ

sin (\frac{3 π}{2} + θ) = - cos θ, cos (\frac{3 π}{2} + θ) = sin θ

6.6 Ratios of $2 π \pm θ$

sin (2 π \pm θ) = \pm sin θ, cos (2 π \pm θ) = cos θ, tan (2 π \pm θ) = \pm tan θ

7. Graphs of Trigonometric Functions

The graphs make the domain, range and periodicity of each function immediately visible.

7.1 Sine and Cosine

Domain $R$ , range $[- 1, 1]$ , period $2 π$ . Cosine is sine shifted left by $π /2$ : $cos x = sin (x + π /2)$ .

Figure 1: Graph of

y = sin x

over

[- 2 π, 2 π]

Figure 2: Graph of

y = cos x

over

[- 2 π, 2 π]

7.2 Tangent and Cotangent

Domain excludes the asymptote points, range $R$ , period $π$ .

Figure 3: Graph of

y = tan x

. Vertical dashed lines are asymptotes at

x = (2 n + 1) π /2

Figure 4: Graph of

y = cot x

. Vertical dashed lines are asymptotes at

x = nπ

7.3 Secant and Cosecant

Range $(- \infty, - 1] \cup [1, \infty)$ ; no value in $(- 1, 1)$ . Period $2 π$ .

Figure 5: Graph of

y = sec x

. Values lie in

(- \infty, - 1] \cup [1, \infty)

Figure 6: Graph of

y = csc x

. Values lie in

(- \infty, - 1] \cup [1, \infty)

8. Domain, Range and Period - Summary

Function	Domain	Range	Period
$sin x$	$R$	$[- 1, 1]$	$2 π$
$cos x$	$R$	$[- 1, 1]$	$2 π$
$tan x$	$R - {(2 n + 1) π /2}$	$R$	$π$
$cot x$	$R - {nπ}$	$R$	$π$
$sec x$	$R - {(2 n + 1) π /2}$	$(- \infty, - 1] \cup [1, \infty)$	$2 π$
$csc x$	$R - {nπ}$	$(- \infty, - 1] \cup [1, \infty)$	$2 π$

9. Periodicity in Detail

A function $f$ is periodic with period $T$ if $f (x + T) = f (x)$ for every $x$ in the domain, and $T$ is the smallest such positive number.

9.1 Periods of Transformed Functions

Period Rules

Period of $sin (a x + b)$ , $cos (a x + b)$ , $csc (a x + b)$ , $sec (a x + b)$ is $\frac{2 π}{∣ a ∣}$ .
Period of $tan (a x + b)$ , $cot (a x + b)$ is $\frac{π}{∣ a ∣}$ .
Period of $∣ sin x ∣$ , $∣ cos x ∣$ , $∣ tan x ∣$ , $∣ cot x ∣$ is $π$ (halved because absolute value folds).
Period of $sin^{n} x$ , $cos^{n} x$ is $π$ if $n$ is even, else $2 π$ .
Period of $f (x) + g (x)$ is the LCM of periods of $f$ and $g$ (when it exists).

Solved Example 2

Find the period of

f (x) = sin 2 x + cos 3 x

Solution:

Period of $sin 2 x$ is $2 π /2 = π$ . Period of $cos 3 x$ is $2 π /3$ .

LCM of $π$ and $2 π /3$ : write as $π = 3 π /3$ , so LCM $= 6 π /3 = 2 π$ . Hence period of $f$ is $2 π$ .

10. Worked Examples

Solved Example 3

Prove that

sin (45° + A) cos (45° - B) + cos (45° + A) sin (45° - B) = cos (A - B)

Solution:

Recognise the pattern $sin X cos Y + cos X sin Y = sin (X + Y)$ with $X = 45° + A$ , $Y = 45° - B$ :

= sin ((45° + A) + (45° - B)) = sin (90° + (A - B)) = cos (A - B) .

Solved Example 4

Prove

tan (\frac{π}{4} + θ) \cdot tan (\frac{3 π}{4} + θ) = - 1

Solution:

$tan (\frac{π}{4} + θ) = \frac{1 + tan θ}{1 - tan θ}$ (using $tan 45° = 1$ ).

$tan (\frac{3 π}{4} + θ) = tan (π - (\frac{π}{4} - θ)) = - tan (\frac{π}{4} - θ) = - \frac{1 - tan θ}{1 + tan θ}$ .

Product $= \frac{1 + tan θ}{1 - tan θ} \cdot (- \frac{1 - tan θ}{1 + tan θ}) = - 1.$

Solved Example 5

sin α = 3/5

and

cos β = 5/13

with

α, β

in the first quadrant, find

sin (α + β)

Solution:

$cos α = 1 - 9/25 = 4/5$ ; $sin β = 1 - 25/169 = 12/13$ .

$sin (α + β) = sin α cos β + cos α sin β = \frac{3}{5} \cdot \frac{5}{13} + \frac{4}{5} \cdot \frac{12}{13} = \frac{15 + 48}{65} = \frac{63}{65}$ .

Solved Example 6

Find the value of

sin 105° \cdot cos 75°

Solution:

$sin 105° = sin (180° - 75°) = sin 75° = cos 15°$ ; $cos 75° = sin 15°$ .

So the product $= cos 15° sin 15° = \frac{1}{2} sin 30° = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$ .

Solved Example 7

Find the domain of

f (x) = cos x - \frac{1}{2}

Solution:

Need $cos x \geq 1/2$ , i.e. $cos x \geq cos (π /3)$ . Since cosine is decreasing on $[0, π]$ and even and periodic, $cos x \geq 1/2$ iff $x \in [- π /3 + 2 nπ, π /3 + 2 nπ]$ for some integer $n$ .

x \in n \in Z ⋃ [2 nπ - \frac{π}{3}, 2 nπ + \frac{π}{3}]

Solved Example 8

Find the period of

f (x) = sin (\frac{x}{2}) + cos (\frac{x}{3})

Solution:

Period of $sin (x /2)$ is $2 π / (1/2) = 4 π$ ; period of $cos (x /3)$ is $2 π / (1/3) = 6 π$ .

LCM of $4 π$ and $6 π$ is $12 π$ . Hence period $= 12 π$ .

Common Mistakes to Avoid

Watch out

Assuming $sin (A + B) = sin A + sin B$ . Sine is not additive. Use the compound angle formula $sin (A + B) = sin A cos B + cos A sin B$ .
Mixing degrees and radians. Formulas like $ℓ = r θ$ require $θ$ in radians. If your $θ$ is in degrees, convert first.
Wrong sign in ASTC. The rule tells you where each function is positive - everywhere else that function is negative. Double-check the quadrant of the given angle.
Forgetting reciprocal restrictions. $csc x$ is undefined wherever $sin x = 0$ , not just at $x = 0$ . Same for $sec x$ at every zero of $cos x$ .
Using $sin^{2} θ = sin θ$ . Correct is $sin^{2} θ = ∣ sin θ ∣$ , which equals $\pm sin θ$ depending on the sign.
Period of $∣ sin x ∣$ is $2 π$ . Wrong - the absolute value reflects the negative half up, halving the period to $π$ .
$tan (90°)$ = huge number. $tan 90°$ is undefined, not a number. Values near $90°$ get arbitrarily large but there is no value at $90°$ .

Frequently Asked Questions

What are the six trigonometric functions?

The six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec or csc). They are defined as ratios of sides of a right triangle for acute angles, and extended to all real angles using the unit circle.

What is the domain and range of sin x and cos x?

Both $sin x$ and $cos x$ have domain $R$ (all real numbers) and range $[- 1, 1]$ . They are the only two trigonometric functions defined for every real angle.

What is the ASTC rule in trigonometry?

The ASTC rule (All, Sine, Tan, Cos) tells you which trigonometric functions are positive in each quadrant. In quadrant I all six functions are positive; in II only sin and cosec are positive; in III only tan and cot; in IV only cos and sec.

Why is tan x not defined at π/2?

Because $tan x = sin x / cos x$ and $cos (π /2) = 0$ , division by zero makes $tan (π /2)$ undefined. The graph shows a vertical asymptote at $x = π /2$ and at every $x = (2 n + 1) π /2$ .

What is the period of trigonometric functions?

$sin x$ , $cos x$ , $csc x$ , and $sec x$ have period $2 π$ . $tan x$ and $cot x$ have period $π$ . This means the graphs repeat exactly after these intervals.

How do you convert degrees to radians?

Use the relation $180° = π$ radians. So degree to radian: multiply by $π /180$ . Radian to degree: multiply by $180/ π$ . For example $60° = 60 \times π /180 = π /3$ radians.

What is 1 radian in degrees?

1 radian $= 180/ π \approx 57.2958°$ . A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.

What are the values of sin, cos, tan at standard angles?

Key values: $sin 0° = 0$ , $sin 30° = 1/2$ , $sin 45° = 1/ 2$ , $sin 60° = 3 /2$ , $sin 90° = 1$ . $cos$ values are the same in reverse order. $tan$ at these angles: $0, 1/ 3, 1, 3,$ undefined.

Previous year questions on Trigonometric Functions

3 questions from past papers, each with a step-by-step solution.

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1. Angle and its Measurement

1.1 Three Systems of Measurement

1.2 Radian - the Natural Unit

1.3 Arc Length and Sector Area

2. Trigonometric Ratios of an Acute Angle

3. Extension to Any Real Angle - the Unit Circle

3.1 Quadrantal Angles

4. Signs of Trigonometric Functions - the ASTC Rule

5. Values at Standard Angles

6. Allied Angles

6.1 Ratios of −θ

6.2 Ratios of π/2−θ (complementary angles)

6.3 Ratios of π/2+θ

6.4 Ratios of π±θ

6.5 Ratios of 3π/2±θ

6.6 Ratios of 2π±θ

7. Graphs of Trigonometric Functions

7.1 Sine and Cosine

7.2 Tangent and Cotangent

7.3 Secant and Cosecant

8. Domain, Range and Period - Summary

9. Periodicity in Detail

9.1 Periods of Transformed Functions

10. Worked Examples

Common Mistakes to Avoid

Frequently Asked Questions

What are the six trigonometric functions?

What is the domain and range of sin x and cos x?

What is the ASTC rule in trigonometry?

Why is tan x not defined at π/2?

What is the period of trigonometric functions?

How do you convert degrees to radians?

What is 1 radian in degrees?

What are the values of sin, cos, tan at standard angles?

Previous year questions on Trigonometric Functions

Ready to master Trigonometry?

6.1 Ratios of $- θ$

6.2 Ratios of $π /2 - θ$ (complementary angles)

6.3 Ratios of $π /2 + θ$

6.4 Ratios of $π \pm θ$

6.5 Ratios of $3 π /2 \pm θ$

6.6 Ratios of $2 π \pm θ$