NCERT Plus One (Class 11) Mathematics • Chapter 3

Trigonometric Functions

Q: Convert 40° 20' into radian measure.

40° 20' = 40 20 60 ° = 121 3 ° . In radians: 121 3 × π 180 = 121π 540 rad.

Q: A wheel makes 360 revolutions in one minute. Through how many radians does it turn in 1 second?

Revolutions per second = 360 / 60 = 6 . 1 revolution = 2π radians ⇒ in 1 s, angle = 6 × 2π = 12π radians .

Q: If tan x = −5/12 and x lies in the second quadrant, find the values of sin x and cos x.

sec²x = 1 + tan²x = 1 + 25/144 = 169/144 ⇒ sec x = ±13/12 . In Q2, cos x 0 : • cos x = −12/13 • sin x = tan x · cos x = (−5/12)(−12/13) = 5/13 .

Q: Find the value of sin(31π/3).

31π 3 = 10π + π 3 = 5(2π) + π 3 . Since period of sine is 2π , sin(31π/3) = sin(π/3) = √3 / 2 .

Q: Find the exact value of sin 75°.

sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = 1 √2 · √3 2 + 1 √2 · 1 2 = √6 + √2 4 .

Q: Prove that tan 3x tan 2x tan x = tan 3x − tan 2x − tan x.

Write 3x = 2x + x ⇒ tan 3x = tan(2x + x) = tan 2x + tan x 1 − tan 2x tan x . Cross-multiply: tan 3x − tan 3x tan 2x tan x = tan 2x + tan x . Rearrange: tan 3x − tan 2x − tan x = tan 3x tan 2x tan x . Proved!

Q: Prove that (cos 9x − cos 5x) / (sin 17x − sin 3x) = −sin 2x / cos 10x.

LHS numerator: −2 sin 7x sin 2x (watch the minus!). LHS denominator: 2 cos 10x sin 7x . Cancel 2 sin 7x → −sin 2x / cos 10x = RHS !

Q: Prove that cos 6x = 32 cos⁶x − 48 cos⁴x + 18 cos²x − 1.

Write cos 6x = cos 2(3x) = 2 cos²(3x) − 1 . Substitute cos 3x = 4 cos³x − 3 cos x : 2(4 cos³x − 3 cos x)² − 1 = 2(16 cos⁶x − 24 cos⁴x + 9 cos²x) − 1 = 32 cos⁶x − 48 cos⁴x + 18 cos²x − 1 .

Radian measure, unit circle & ASTC signs, 6 trig graphs, sum/difference, double/triple angles & C–D formulas.

📄 Official NCERT Textbook PDF

Topic 3.2: Degree vs Radian Measure & Arc Formula (l = rθ)

Picture slicing a giant Kerala birthday pizza! Dividing a full circle into 360 tiny slivers gives Degrees (1° = 60', 1' = 60''). But mathematicians prefer a natural yardstick: wrap a piece of string equal to the radius r along the crust! The angle inside that slice is exactly 1 Radian (≈ 57.3°), making arc length effortless: l = rθ!

GOLDEN BRIDGEπ radians = 180° | 2π rad = 360°

CONVERSIONSDeg → Rad: multiply by π/180
Rad → Deg: multiply by 180/π

ARC LENGTHl = r · θ (θ MUST be in radians!)

🧠 Memory Trick: Want π inside the answer (Radians)? Multiply by π/180! Want to cancel π out (Degrees)? Multiply by 180/π!

💡 Simple Kid-Check: A full circle circumference is 2πr. Divide by radius r and you get 2π radians! That's why half a turn (180°) equals π radians, and a quarter-turn (90°) equals π/2!

Degrees	30°	45°	60°	90°	180°	270°	360°
Radians	`π/6`	`π/4`	`π/3`	`π/2`	`π`	`3π/2`	`2π`

🎡 Quick Example: A giant wheel of radius r = 14 m turns by 60°. Find the arc travelled!
Step 1: Convert 60° to radians → θ = 60 × π180 = π3 rad.
Step 2: l = rθ = 14 × 227×3 = 443 ≈ 14.67 m!

NEVER plug degrees straight into l = rθ! Writing l = 14 × 60 = 840 m is a classic DHSE 2-mark blunder! Always convert degrees to radians FIRST using π/180! Also remember 1° = 60' (minutes) and 1' = 60'' (seconds) — NOT 100!

Topic 3.3: Unit Circle Definition & ASTC Quadrant Signs

Imagine a glowing ferris wheel of radius 1! As your seat spins round the circle at angle x, your horizontal shadow is cos x and your vertical height is sin x. Depending on which of the four sky-quadrants you enter, cos x and sin x flip between positive and negative following one catchy phrase: "All Students Take Coffee" (ASTC)!

UNIT CIRCLE (r = 1)Point at angle x is (cos x, sin x)! So cos²x + sin²x = 1

ASTC QUADRANTSQ1: All + | Q2: Sin/cosec +
Q3: Tan/cot + | Q4: Cos/sec +

NEGATIVE ANGLEScos(−x) = cos x (eats minus!)
sin(−x) = −sin x, tan(−x) = −tan x

🧠 Memory Trick: A-S-T-C ("Add Sugar To Coffee") marks who stays POSITIVE in Quadrants I, II, III, IV! And for allied angles: 90°/270° FLIPS ratio (sin ↔ cos), 180°/360° KEEPS ratio same!

💡 Simple Kid-Check: Right of the Y-axis means cos x > 0 (Q1 & Q4). Above the X-axis means sin x > 0 (Q1 & Q2). Divide them for tan x = sin x / cos x: positive only when both share the same sign (Q1 & Q3)!

1
Three Pythagorean Identities: sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = cosec²x.
2
Allied Angle 2-Step Rule: First check if angle uses π/2 or 3π/2 (odd 90° → switch sin↔cos, tan↔cot, sec↔cosec) vs π, 2π (keep same!).
3
Check Sign Using ORIGINAL Function's Quadrant: E.g. cos(90° + x) lies in Q2 where cos is negative → −sin x!

Huge allied-angle mistake: Checking ASTC sign using the NEW ratio instead of the ORIGINAL ratio! For cos(π/2 + x), Q2 has sin > 0, so kids mistakenly write +sin x! WRONG — the original function is cos, which is negative in Q2, so cos(π/2 + x) = −sin x!

Topic 3.3: Domain, Range & Periodic Graphs of 6 Trig Functions

Watch ocean waves at Kovalam beach rising and falling in smooth identical cycles! Both sin x and cos x stay forever trapped inside the safe elevator between −1 and +1, repeating every 2π. Meanwhile, tan x and sec x zoom to ±∞ every time cos x hits 0!

BOUNDED PAIRsin x, cos x: Domain ℝ, Range [−1, 1], Period 2π

UNBOUNDED PAIRtan x, cot x: Range ℝ, Period π (half!)

RECIPROCAL GAPcosec x, sec x: Range ℝ − (−1, 1) (never inside (−1,1)!)

🧠 Memory Trick: Where does division by zero create vertical asymptote walls? Whenever cos x = 0 (odd π/2: bans tan, sec) or sin x = 0 (integer nπ: bans cot, cosec)!

💡 Simple Kid-Check: Since sin x never exceeds 1, its flip cosec x = 1/sin x can NEVER fall between −1 and +1! Either cosec x ≥ 1 or cosec x ≤ −1!

Function	Domain	Range	Period
`sin x`	`ℝ`	`[−1, 1]`	`2π`
`cos x`	`ℝ`	`[−1, 1]`	`2π`
`tan x`	`ℝ − {(2n+1)π/2 : n ∈ ℤ}`	`ℝ`	`π`
`cot x`	`ℝ − {nπ : n ∈ ℤ}`	`ℝ`	`π`
`cosec x`	`ℝ − {nπ : n ∈ ℤ}`	`ℝ − (−1, 1)`	`2π`
`sec x`	`ℝ − {(2n+1)π/2 : n ∈ ℤ}`	`ℝ − (−1, 1)`	`2π`

Watch out for impossible MCQ equations like "Find x if sec x = 1/2" or "sin x = 5/4"! Both have NO REAL SOLUTION because |sin x| ≤ 1 and |sec x| ≥ 1! Also remember tan x and cot x repeat every π, NOT 2π!

Topic 3.4: Sum & Difference Identities (cos(x±y), sin(x±y), tan(x±y))

How can you find sin 75° or cos 15° WITHOUT a calculator? Split 75° = 45° + 30° or 15° = 45° − 30°! Just NEVER fall into the rookie trap of writing sin(45° + 30°) = sin 45° + sin 30° — trigonometric functions do NOT distribute like regular brackets; they obey special Sum & Difference Formulas!

COSINE (SELFISH!)cos(x ± y) = cos x cos y ∓ sin x sin y
(Keeps cos together, FLIPS sign!)

SINE (FRIENDLY!)sin(x ± y) = sin x cos y ± cos x sin y
(Mixes sin & cos, KEEPS sign!)

TANGENT FRACTIONtan(x ± y) = tan x ± tan y1 ∓ tan x tan y

🧠 Memory Trick: Sine shares & stays honest (mixes sin/cos, same sign); Cosine is selfish & opposite (cos cos together, opposite sign)!

💡 Simple Kid-Check: Test why sin(x + y) ≠ sin x + sin y using 30° and 60°: sin(30° + 60°) = sin 90° = 1, whereas sin 30° + sin 60° = 0.5 + 0.866 = 1.366 (impossible, since sine never beats 1!).

Super-Handy Exam Shortcut: tan(π/4 + x) = 1 + tan x1 − tan x | tan(π/4 − x) = 1 − tan x1 + tan x

🧮 Classic Value (sin 15° / cos 75°):
sin 15° = sin(45° − 30°) = sin 45° cos 30° − cos 45° sin 30° = 1√2·√32 − 1√2·12 = √3 − 12√2 (or √6 − √24)!

Top sign trap: In cos(x + y), the middle sign is MINUS (cos x cos y − sin x sin y), and in cos(x − y), the middle sign is PLUS! Also in tan(x + y), top keeps + and bottom flips to −!

Topic 3.4: Double & Triple Angle Formulae (sin 2x, cos 2x, tan 2x, sin 3x)

Set y = x inside your Sum formulas — boom, x + x = 2x gives every Double Angle Formula automatically! Need Triple Angles (3x)? Just use 2x + x! These power-houses let engineers halve or triple wave frequencies in AC circuits, sound synthesis, and calculus!

SIN 2x & TAN 2xsin 2x = 2 sin x cos x = 2 tan x1 + tan²x
tan 2x = 2 tan x1 − tan²x

COS 2x (4 FACES!)cos²x − sin²x = 2cos²x − 1 = 1 − 2sin²x = 1 − tan²x1 + tan²x

TRIPLE ANGLESsin 3x = 3 sin x − 4 sin³x
cos 3x = 4 cos³x − 3 cos x

🧠 Memory Trick: Watch the denominator of sin 2x vs tan 2x: Sine has PLUS (1 + tan²x), Tan has MINUS (1 − tan²x)! And for triple angles: 34 for Sine (3s − 4s³), 43 for Cosine (4c³ − 3c)!

💡 Simple Kid-Check: Rearrange cos 2x to get the two MOST-USED formulas in Class 12 integration:
1 + cos 2x = 2 cos²x (plus likes cos!) and 1 − cos 2x = 2 sin²x (minus likes sin!)

1
Tangent Triple Formula: tan 3x = 3 tan x − tan³x1 − 3 tan²x.
2
Half-Angle Magic: Replace x → x/2 everywhere! E.g. 1 − cos x = 2 sin²(x/2) and sin x = 2 sin(x/2) cos(x/2).

⚡ Quick Proof: Simplify 1 − cos 2xsin 2x = 2 sin²x2 sin x cos x = tan x in two lines!

Two super-common traps: (1) Putting 1 − tan²x in the denominator of sin 2x instead of 1 + tan²x! Remember sin 2x ALWAYS stays bounded inside [−1, 1], so its denominator 1 + tan²x can NEVER become zero! (2) Swapping 34 and 43 in sin 3x vs cos 3x!

Topic 3.4: Product-to-Sum & Sum-to-Product (C–D) Transformations

When two musical notes of slightly different frequencies C and D mix in air, they create pulsing volume beats! In Maths, turning a SUM like sin C + sin D into a clean PRODUCT (so common factors cancel!) uses the iconic C–D Formulas — and reversing them turns products back into sums!

SINE C–D RULESsin C + sin D = 2 sin(C+D2) cos(C−D2)
sin C − sin D = 2 cos(C+D2) sin(C−D2)

COSINE C–D RULEScos C + cos D = 2 cos(C+D2) cos(C−D2)
cos C − cos D = −2 sin(C+D2) sin(C−D2)

PRODUCT → SUM2 sin x cos y = sin(x+y) + sin(x−y)
2 cos x cos y = cos(x+y) + cos(x−y)

🧠 Memory Trick: Watch the FOUR C–D patterns: S+S → 2SC, S−S → 2CS, C+C → 2CC, and C−C → −2SS (that MINUS in front of 2SS is pure gold!)

💡 Simple Kid-Check: Inside every C–D formula, the first angle is ALWAYS the average C + D2 and the second angle is ALWAYS half-difference C − D2! Only the trig names and that one minus sign change!

Complete Product-to-Sum Quartet:
1) 2 sin x cos y = sin(x + y) + sin(x − y)
2) 2 cos x sin y = sin(x + y) − sin(x − y)
3) 2 cos x cos y = cos(x + y) + cos(x − y)
4) 2 sin x sin y = cos(x − y) − cos(x + y) (note cos(x−y) comes FIRST!)

🏆 Classic 3-Mark DHSE Proof: Simplify sin 5x + sin 3xcos 5x + cos 3x:
Top = 2 sin 4x cos x, Bottom = 2 cos 4x cos x → cancel 2 cos x → tan 4x instantly!

The #1 mistake in Kerala HSE 3-mark and 4-mark proofs: forgetting the minus sign in cos C − cos D = −2 sin(C+D2) sin(C−D2) (unless you flip the second bracket to (D − C)/2)! And in Product-to-Sum, remember 2 sin x sin y = cos(x − y) − cos(x + y) puts (x − y) FIRST!

Solved Previous Year Questions (Kerala DHSE & NCERT)

Q1 (Kerala HSE March • 1 Marks): Convert 40° 20' into radian measure.

40° 20' = 402060° = 1213°.
In radians: 1213 × π180 = 121π540 rad.

Q2 (Kerala HSE Say • 2 Marks): A wheel makes 360 revolutions in one minute. Through how many radians does it turn in 1 second?

Revolutions per second = 360 / 60 = 6.
1 revolution = 2π radians ⇒ in 1 s, angle = 6 × 2π = 12π radians.

Q3 (Kerala HSE March • 3 Marks): If tan x = −5/12 and x lies in the second quadrant, find the values of sin x and cos x.

sec²x = 1 + tan²x = 1 + 25/144 = 169/144 ⇒ sec x = ±13/12.
In Q2, cos x < 0 and sin x > 0:
• cos x = −12/13
• sin x = tan x · cos x = (−5/12)(−12/13) = 5/13.

Q4 (Kerala HSE March • 2 Marks): Find the value of sin(31π/3).

31π3 = 10π + π3 = 5(2π) + π3.
Since period of sine is 2π, sin(31π/3) = sin(π/3) = √3 / 2.

Q5 (Kerala HSE Model • 2 Marks): Find the exact value of sin 75°.

sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = 1√2·√32 + 1√2·12 = √6 + √24.

Q6 (Kerala HSE March • 3 Marks): Prove that tan 3x tan 2x tan x = tan 3x − tan 2x − tan x.

Write 3x = 2x + x ⇒ tan 3x = tan(2x + x) = tan 2x + tan x1 − tan 2x tan x.
Cross-multiply: tan 3x − tan 3x tan 2x tan x = tan 2x + tan x.
Rearrange: tan 3x − tan 2x − tan x = tan 3x tan 2x tan x. Proved!

Q7 (Kerala HSE Say • 3 Marks): Prove that (cos 9x − cos 5x) / (sin 17x − sin 3x) = −sin 2x / cos 10x.

LHS numerator: −2 sin 7x sin 2x (watch the minus!).
LHS denominator: 2 cos 10x sin 7x.
Cancel 2 sin 7x → −sin 2x / cos 10x = RHS!

Q8 (NCERT / DHSE Model • 4 Marks): Prove that cos 6x = 32 cos⁶x − 48 cos⁴x + 18 cos²x − 1.

Write cos 6x = cos 2(3x) = 2 cos²(3x) − 1.
Substitute cos 3x = 4 cos³x − 3 cos x:

2(4 cos³x − 3 cos x)² − 1 = 2(16 cos⁶x − 24 cos⁴x + 9 cos²x) − 1 = 32 cos⁶x − 48 cos⁴x + 18 cos²x − 1