1. Chapter at a Glance
In Class 10, trigonometric ratios were defined only for acute angles of a right triangle. But a wheel turns through 400°, a pendulum swings to negative angles, and waves repeat forever. To describe all of these, we extend sine, cosine and the other ratios to every real number using the unit circle, and we measure angles in radians as well as degrees. This chapter develops radian measure, the six trigonometric functions with their signs, domains, ranges and graphs, and the powerful sum, difference, multiple-angle and transformation formulas — essential for calculus and heavily tested in JEE.
- Positive and negative angles; degree and radian measure; conversion.
- Arc length l = rθ.
- Trigonometric functions of any real number via the unit circle.
- Signs in the four quadrants; domain, range, period and graphs.
- Formulas for sin(x ± y), cos(x ± y), tan(x ± y).
- Allied angles (π/2 ± x, π ± x, 2π ± x).
- Double and triple angle formulas; sum-to-product and product-to-sum formulas.
2. Measuring Angles
An angle is the amount of rotation of a ray from its initial side to its terminal side: anticlockwise = positive, clockwise = negative.
One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. A full turn = 2π radians = 360°.
π radians = 180° → 1 rad ≈ 57°16′; 1° = π/180 rad ≈ 0.01746 rad.
Arc length: l = rθ (θ in radians).
| Degrees | 30° | 45° | 60° | 90° | 120° | 135° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|---|
| Radians | π/6 | π/4 | π/3 | π/2 | 2π/3 | 3π/4 | π | 3π/2 | 2π |
(a) Convert 40°20′ to radians. (b) A wheel makes 360 revolutions per minute. Through how many radians does it turn in one second?
Solution: (a) 40⅓° × π/180 = 121π/540 rad. (b) 6 rev/s × 2π = 12π rad.
Find the angle in radians subtended at the centre by an arc of length 22 cm in a circle of radius 100 cm, and in degrees (π = 22/7).
Solution: θ = 22/100 = 0.22 rad = 0.22 × 180/π ≈ 12.6°.
3. Trigonometric Functions of Any Angle
Draw the unit circle (radius 1, centre O). The terminal side of angle x meets it at P(a, b). Define cos x = a and sin x = b. Then tan x = sin x/cos x, cot x = cos x/sin x, sec x = 1/cos x, cosec x = 1/sin x (where defined).
- sin²x + cos²x = 1 for all x; 1 + tan²x = sec²x; 1 + cot²x = cosec²x.
- −1 ≤ sin x ≤ 1, −1 ≤ cos x ≤ 1.
- sin x = 0 ⇔ x = nπ; cos x = 0 ⇔ x = (2n + 1)π/2 (n ∈ ℤ).
- sin(−x) = −sin x (odd); cos(−x) = cos x (even).
- sin(2nπ + x) = sin x, cos(2nπ + x) = cos x (period 2π).
| Function | Domain | Range | Period |
|---|---|---|---|
| sin x | ℝ | [−1, 1] | 2π |
| cos x | ℝ | [−1, 1] | 2π |
| tan x | ℝ − {(2n + 1)π/2} | ℝ | π |
| cot x | ℝ − {nπ} | ℝ | π |
| sec x | ℝ − {(2n + 1)π/2} | (−∞, −1] ∪ [1, ∞) | 2π |
| cosec x | ℝ − {nπ} | (−∞, −1] ∪ [1, ∞) | 2π |
If cos x = −3/5 and x lies in the third quadrant, find the other five functions.
Solution: sin x = −4/5 (negative in III). tan x = 4/3, cot x = 3/4, sec x = −5/3, cosec x = −5/4.
Find sin(31π/3) and cos(−1710°).
Solution: 31π/3 = 10π + π/3 → sin = sin π/3 = √3/2. cos(−1710°) = cos 1710° = cos(4 × 360° + 270°) = cos 270° = 0.
4. Sum and Difference Formulas
- cos(x + y) = cos x cos y − sin x sin y
- cos(x − y) = cos x cos y + sin x sin y
- sin(x + y) = sin x cos y + cos x sin y
- sin(x − y) = sin x cos y − cos x sin y
- tan(x ± y) = (tan x ± tan y)/(1 ∓ tan x tan y)
- cot(x ± y) = (cot x cot y ∓ 1)/(cot y ± cot x)
The formula for cos(x + y) is proved on the unit circle by equating the lengths of two equal chords; all the others follow from it.
Find sin 15° and tan 75°.
Solution: sin(45° − 30°) = (1/√2)(√3/2) − (1/√2)(1/2) = (√3 − 1)/(2√2). tan(45° + 30°) = (1 + 1/√3)/(1 − 1/√3) = (√3 + 1)/(√3 − 1) = 2 + √3.
Allied angles
| Angle | sin | cos | tan |
|---|---|---|---|
| π/2 − x | cos x | sin x | cot x |
| π/2 + x | cos x | −sin x | −cot x |
| π − x | sin x | −cos x | −tan x |
| π + x | −sin x | −cos x | tan x |
| 2π − x | −sin x | cos x | −tan x |
Rule: with π/2 or 3π/2 the function changes (sin ↔ cos, tan ↔ cot); with π or 2π it stays. The sign is that of the original function in the quadrant where the angle lies (treating x as acute).
5. Multiple-angle Formulas
- sin 2x = 2 sin x cos x = 2 tan x/(1 + tan²x)
- cos 2x = cos²x − sin²x = 2cos²x − 1 = 1 − 2sin²x = (1 − tan²x)/(1 + tan²x)
- tan 2x = 2 tan x/(1 − tan²x)
- sin 3x = 3 sin x − 4 sin³x
- cos 3x = 4 cos³x − 3 cos x
- tan 3x = (3 tan x − tan³x)/(1 − 3 tan²x)
Useful rearrangements: cos²x = (1 + cos 2x)/2, sin²x = (1 − cos 2x)/2.
Find sin(x/2), cos(x/2), tan(x/2) if tan x = −4/3 and x is in quadrant II.
Solution: cos x = −3/5. x/2 is in quadrant I. sin²(x/2) = (1 + 3/5)/2 = 4/5 → sin(x/2) = 2/√5. cos²(x/2) = (1 − 3/5)/2 = 1/5 → cos(x/2) = 1/√5. tan(x/2) = 2.
6. Transformation Formulas
- cos x + cos y = 2 cos((x + y)/2) cos((x − y)/2)
- cos x − cos y = −2 sin((x + y)/2) sin((x − y)/2)
- sin x + sin y = 2 sin((x + y)/2) cos((x − y)/2)
- sin x − sin y = 2 cos((x + y)/2) sin((x − y)/2)
- 2 cos x cos y = cos(x + y) + cos(x − y)
- −2 sin x sin y = cos(x + y) − cos(x − y)
- 2 sin x cos y = sin(x + y) + sin(x − y)
- 2 cos x sin y = sin(x + y) − sin(x − y)
Prove (cos 7x + cos 5x)/(sin 7x − sin 5x) = cot x.
Proof: Numerator = 2 cos 6x cos x; denominator = 2 cos 6x sin x. Ratio = cos x/sin x = cot x. ∎
7. Solved Problems (Board Pattern)
Prove: sin²(π/6) + cos²(π/3) − tan²(π/4) = −1/2.
Solution: ¼ + ¼ − 1 = −½. ∎
Prove: cos(π/4 − x) cos(π/4 − y) − sin(π/4 − x) sin(π/4 − y) = sin(x + y).
Solution: LHS = cos[(π/4 − x) + (π/4 − y)] = cos(π/2 − (x + y)) = sin(x + y). ∎
Prove: cos 6x = 32cos⁶x − 48cos⁴x + 18cos²x − 1.
Solution: cos 6x = cos 2(3x) = 2cos²3x − 1 = 2(4cos³x − 3cos x)² − 1 = 2(16cos⁶x − 24cos⁴x + 9cos²x) − 1 = 32cos⁶x − 48cos⁴x + 18cos²x − 1. ∎
8. Master Formula Table
| Topic | Formula |
|---|---|
| Conversion | radians = degrees × π/180 |
| Arc | l = rθ |
| Pythagorean | sin²x + cos²x = 1 |
| cos(x ± y) | cos x cos y ∓ sin x sin y |
| sin(x ± y) | sin x cos y ± cos x sin y |
| tan(x ± y) | (tan x ± tan y)/(1 ∓ tan x tan y) |
| Double angle | sin 2x = 2 sin x cos x; cos 2x = 1 − 2sin²x |
| Triple angle | sin 3x = 3sin x − 4sin³x; cos 3x = 4cos³x − 3cos x |
| Sum to product | sin x + sin y = 2 sin((x+y)/2) cos((x−y)/2) |
| Product to sum | 2 sin x cos y = sin(x + y) + sin(x − y) |
9. Master Summary Table
| Concept | Key idea | Exam focus |
|---|---|---|
| Radian measure | π = 180° | Conversions, l = rθ |
| Unit circle | P = (cos x, sin x) | Signs, values of any angle |
| Periodicity | sin, cos: 2π; tan, cot: π | Reduce large angles |
| Compound angles | cos(x + y) is the base formula | sin 15°, tan 75° |
| Multiple angles | Double/triple | Half-angle values, identities |
| Transformations | Sum ↔ product | Simplify ratios of sums |
10. Mind Map
- Degree, radian
- l = rθ
- Unit circle
- Signs, graphs, periods
- x ± y
- Allied angles
- 2x, 3x
- Sum ↔ product
11. Tips and Tricks
- "All Silver Tea Cups": positive functions in quadrants I, II, III, IV.
- Reduce any angle: remove multiples of 2π (or 360°), then use allied-angle rules.
- When you see sums like cos A + cos B in a fraction, convert to products — common factors usually cancel.
- Using degrees in l = rθ — θ must be in radians.
- sin 2x ≠ 2 sin x; sin(x + y) ≠ sin x + sin y.
- Choosing the wrong sign of a half-angle — decide the quadrant of x/2 first.
12. Important One-liners
- π radians = 180°.
- On the unit circle, cos x is the x-coordinate and sin x the y-coordinate.
- sin is odd and cos is even.
- tan and cot have period π; the others have period 2π.
- sin 15° = (√3 − 1)/(2√2), tan 75° = 2 + √3.
- cos 2x = 2cos²x − 1 = 1 − 2sin²x.
13. Exam-pattern Questions with Detailed Answers
The radian measure of −47°30′ is:
- −19π/72
- −19π/36
- −47π/180
- −π/4
Show answer
−47.5 × π/180 = −95π/360 = −19π/72.
Answer: (A)
Find the value of tan(19π/3).
Show answer
19π/3 = 6π + π/3 → tan(π/3) = √3.
Answer: √3
Convert 5π/3 radians to degrees.
Show answer
5π/3 × 180/π = 300°.
Answer: 300°
If sin x = 3/5 and x lies in quadrant II, find the other five functions.
Show answer
cos x = −4/5; tan x = −3/4; cot x = −4/3; sec x = −5/4; cosec x = 5/3.
Answer: As above
Find cos 105°.
Show answer
cos(60° + 45°) = (½)(1/√2) − (√3/2)(1/√2) = (1 − √3)/(2√2).
Answer: (1 − √3)/(2√2)
Prove: sin(n + 1)x sin(n + 2)x + cos(n + 1)x cos(n + 2)x = cos x.
Show answer
LHS = cos[(n + 2)x − (n + 1)x] = cos x. ∎
Answer: Proved
A pendulum 75 cm long swings through an arc of 21 cm. Find the angle (in degrees) through which it swings (π = 22/7).
Show answer
θ = 21/75 = 0.28 rad = 0.28 × 180 × 7/22 ≈ 16.04°.
Answer: ≈ 16°
Prove: (sin 5x + sin 3x)/(cos 5x + cos 3x) = tan 4x.
Show answer
= (2 sin 4x cos x)/(2 cos 4x cos x) = tan 4x. ∎
Answer: Proved
Prove: cos²x + cos²(x + π/3) + cos²(x − π/3) = 3/2.
Show answer
Use cos²θ = (1 + cos 2θ)/2: sum = 3/2 + ½[cos 2x + cos(2x + 2π/3) + cos(2x − 2π/3)]. The last two = 2 cos 2x cos(2π/3) = −cos 2x. Bracket = 0, so the sum = 3/2. ∎
Answer: Proved
Prove: tan 3x tan 2x tan x = tan 3x − tan 2x − tan x.
Show answer
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. ∎
Answer: Proved
Prove: sin 2x + 2 sin 4x + sin 6x = 4 cos²x sin 4x.
Show answer
sin 6x + sin 2x = 2 sin 4x cos 2x. LHS = 2 sin 4x (cos 2x + 1) = 2 sin 4x × 2cos²x = 4cos²x sin 4x. ∎
Answer: Proved
Find the value of sin 18° (hint: let x = 18°, so 2x = 90° − 3x).
Show answer
sin 2x = cos 3x → 2 sin x cos x = 4cos³x − 3cos x. Divide by cos x: 2 sin x = 4cos²x − 3 = 1 − 4sin²x → 4s² + 2s − 1 = 0 → s = (−1 + √5)/4 (positive root).
Answer: sin 18° = (√5 − 1)/4
14. Quick Revision Summary
- π rad = 180°; l = rθ (θ in radians).
- Unit circle: P = (cos x, sin x); signs by quadrant; periods 2π (sin, cos) and π (tan, cot).
- cos(x ± y), sin(x ± y), tan(x ± y); allied angles.
- sin 2x, cos 2x, tan 2x; sin 3x, cos 3x, tan 3x.
- Sum ↔ product formulas for simplifying and proving identities.