Trigonometric Functions

2.1 Core concepts

Angles and rotation. An angle is a measure of rotation of a ray about its initial point; rotation anticlockwise is positive and clockwise is negative (NCERT §3.2, p. 44). The two arms are the initial side and terminal side, meeting at the vertex.

Degree measure. A degree is 1/360 of one complete revolution; 1° = 60′ and 1′ = 60″. Angles 360°, 270°, 420°, −30°, −420° are illustrated in Fig 3.3 (NCERT §3.2.1, p. 44). Angles greater than 360° correspond to more than one complete rotation.

Radian measure. One radian is the angle subtended at the centre of a unit circle by an arc of length 1 unit; one complete revolution = 2π radian (NCERT §3.2.2, p. 45). For a circle of radius r, an arc of length l subtends an angle θ radian where θ = l/r, giving l = r θ (the most-tested formula in this chapter).

Wrapping the real line. Radian measures and real numbers correspond one-to-one via wrapping the tangent line at A around the unit circle (NCERT §3.2.3, p. 46). This is why trig functions accept any real input.

Conversion. 2π radian = 360°, hence π radian = 180°; 1 radian ≈ 57°16′ and 1° ≈ 0.01746 radian (NCERT §3.2.4, p. 46). Standard table to memorise: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π.

Unit-circle definitions. For a point P(a, b) on the unit circle with ∠AOP = x radian, cos x = a and sin x = b, so a² + b² = 1 gives cos²x + sin²x = 1 (NCERT §3.3, p. 49). The quadrantal values follow at once:

• cos 0 = 1, sin 0 = 0
• cos π/2 = 0, sin π/2 = 1
• cos π = −1, sin π = 0
• cos 3π/2 = 0, sin 3π/2 = −1
• cos 2π = 1, sin 2π = 0 (NCERT §3.3, p. 49–50). Periodicity and zeros. sin(2nπ + x) = sin x and cos(2nπ + x) = cos x for any integer n. sin x = 0 ⇔ x = nπ; cos x = 0 ⇔ x = (2n+1)π/2 (NCERT §3.3, p. 50). Other four functions. cosec x = 1/sin x (x ≠ nπ); sec x = 1/cos x (x ≠ (2n+1)π/2); tan x = sin x/cos x; cot x = cos x/sin x (NCERT §3.3, p. 50). Pythagorean identities. From cos²x + sin²x = 1, dividing by cos²x gives 1 + tan²x = sec²x; dividing by sin²x gives 1 + cot²x = cosec²x (NCERT §3.3, p. 51). Sign by quadrant (ASTC). All positive in Q-I; only sin (and cosec) positive in Q-II; only tan (and cot) positive in Q-III; only cos (and sec) positive in Q-IV (NCERT §3.3.1, p. 52). Mnemonic "All Students Take Coffee". Range, domain, period. Since −1 ≤ a, b ≤ 1, range of sin x and cos x is [−1, 1] for all real x. Domains: sin, cos — all R; tan, sec — R minus odd multiples of π/2; cot, cosec — R minus integral multiples of π (NCERT §3.3.2, p. 52–53). Period of sin, cos, sec, cosec is 2π; period of tan, cot is π (NCERT §3.3.2, p. 54). Sum/difference identities. From the unit-circle congruence P₁OP₃ ≅ P₂OP₄:
• cos(x + y) = cos x cos y − sin x sin y
• cos(x − y) = cos x cos y + sin x sin y (NCERT §3.4, p. 58–59)
• sin(x ± y) = sin x cos y ± cos x sin y (NCERT §3.4, p. 59)
• tan(x ± y) = (tan x ± tan y)/(1 ∓ tan x tan y) (NCERT §3.4, p. 60)
• cot(x ± y) similar (NCERT §3.4, p. 61) Double-angle. cos 2x = cos²x − sin²x = 2cos²x − 1 = 1 − 2sin²x = (1 − tan²x)/(1 + tan²x); sin 2x = 2 sin x cos x = 2 tan x/(1 + tan²x); tan 2x = 2 tan x/(1 − tan²x) (NCERT §3.4, p. 61–62). Triple-angle. sin 3x = 3 sin x − 4 sin³x; cos 3x = 4 cos³x − 3 cos x; tan 3x = (3 tan x − tan³x)/(1 − 3 tan²x) (NCERT §3.4, p. 62). Notable historical context. Trigonometry, literally "triangle measurement", was developed by Hipparchus (~150 BCE), Indian astronomer-mathematicians Aryabhata (~500 CE) and Bhaskara II (~1150 CE), and Arab scholars al-Khwarizmi and al-Battani. The modern function-based formulation, generalising ratios in right triangles to functions on all real numbers, became standard in the 17th century with Euler's introduction of analytic trigonometry. Standard values table. Memorise sin/cos/tan at 0, π/6, π/4, π/3, π/2. Specifically: sin 0 = 0, sin(π/6) = 1/2, sin(π/4) = 1/√2, sin(π/3) = √3/2, sin(π/2) = 1; cos values reversed; tan(π/4) = 1, tan(π/3) = √3, tan(π/6) = 1/√3; tan(π/2) undefined. Sum-to-product / product-to-sum.
• 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) (NCERT §3.4, p. 63)
• 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) (NCERT §3.4, p. 64).

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig 3.1, p. 44 — initial side, terminal side, vertex; sign of angle by direction.
• Fig 3.3, p. 44 — pictorial 360°, 180°, 270°, 420°, −30°, −420°.
• Fig 3.4(i)–(iv), p. 45 — angles of 1 rad, −1 rad, 1½ rad, −1½ rad.
• Fig 3.5, p. 46 — wrapping of real line around unit circle.
• Fig 3.6, p. 49 — defines P(a, b), cos x = a, sin x = b.
• Fig 3.7, p. 51 — symmetry leading to cos(−x) = cos x, sin(−x) = −sin x.
• Figs 3.8–3.13, p. 54–55 — graphs of sin, cos, tan, cot, sec, cosec showing periods and asymptotes.
• Fig 3.14, p. 58 — four points P₁, P₂, P₃, P₄ used in the cos(x + y) congruence proof. Process for sign by quadrant. Step 1 — reduce angle to its equivalent in [0, 2π) by subtracting 2π or 360° appropriately. Step 2 — identify the quadrant. Step 3 — apply ASTC to fix sign. Step 4 — compute magnitude using the reference angle. Process for sum-to-product simplification. Step 1 — match the expression to one of the four standard forms. Step 2 — identify x and y. Step 3 — write the product form with (x + y)/2 and (x − y)/2. Process for general value computation. Step 1 — subtract integer multiples of period (2π for sin/cos, π for tan/cot). Step 2 — reduce to reference angle in [0, π/2]. Step 3 — apply ASTC sign. Step 4 — read off from the standard 30°/45°/60° table. Quick C-D Identity recall. sin C + sin D = 2 sin((C+D)/2) cos((C−D)/2); sin C − sin D = 2 cos((C+D)/2) sin((C−D)/2). cos C + cos D = 2 cos((C+D)/2) cos((C−D)/2); cos C − cos D = −2 sin((C+D)/2) sin((C−D)/2). Notice only "cos − cos" carries the minus sign.

2.4 Common confusions / NTA trap points

• Degree vs radian. sin 30 (interpreted as 30 radians) ≠ sin 30°. Bare θ = radians (NCERT convention, p. 47).
• l = r θ with θ in degrees. Formula valid only with θ in radians; failing to convert produces a × 180/π factor error.
• Sign in quadrant. cos x is negative in Q-II, tan x negative in Q-IV — the ASTC table on p. 52 is non-negotiable.
• Three forms of cos 2x. cos²x − sin²x, 2 cos²x − 1, 1 − 2 sin²x — NTA shuffles them as distractors.
• Domain exclusion. tan x and sec x undefined at odd multiples of π/2; cosec x and cot x undefined at integer multiples of π.
• Sign in cos − cos. cos x − cos y = −2 sin((x+y)/2) sin((x−y)/2); students drop the minus.
• Period of sec / cosec. Both are 2π, not π.
• Triple angle sign. sin 3x = 3 sin x − 4 sin³x but cos 3x = 4 cos³x − 3 cos x — the "3" and "4" swap positions.
• Domain of √(sin x). Requires sin x ≥ 0, i.e. x ∈ [2nπ, 2nπ + π].
• Sec range. (−∞, −1] ∪ [1, ∞), not [−1, 1].
• cot 0 undefined, but tan 0 = 0 — easy to swap.
• General solutions. sin x = sin α ⇒ x = nπ + (−1)ⁿα is examined in Class-XII chapters, but the seed periodicity is from Class XI here.
• Sin and cos symmetry. sin is odd (graph symmetric about origin) and cos is even (symmetric about y-axis). Forgetting this leads to wrong signs in identity reductions.
• Confusing degree minutes with decimal degrees. 30°30′ is 30.5°, not 30.30°; minutes are 1/60 of a degree.
• Treating tan as continuous at π/2. tan is undefined at odd multiples of π/2; the graph has vertical asymptotes there.
• Wrong choice of reference angle. The reference angle is the acute angle between the terminal side and the x-axis (not the y-axis). Mis-identifying it flips the value.
• Half-angle ambiguity. sin(x/2) = ±√((1 − cos x)/2); the sign depends on the quadrant of x/2, not of x.
• Misreading product-to-sum. The identity 2 sin A cos B = sin(A + B) + sin(A − B) preserves sin on RHS only if the first factor is sin; with cos sin, it gives sin(A + B) − sin(A − B).
• Forgetting that cosec, sec values lie in (−∞, −1] ∪ [1, ∞). Never expect cosec x = 0.5 — outside the range.
• Confusing radian and degree conversion direction. Multiply degrees by π/180 to get radians; multiply radians by 180/π to get degrees.

2.5 Key formulas & theorems

2.6 Solved examples

Example 1 — Arc length. In a circle of radius 100 cm, an arc subtends an angle 30° at the centre. Find the arc length. Step 1 — Convert: 30° = 30 × π/180 = π/6 rad. Step 2 — l = r θ = 100 × π/6 = 50π/3 cm. Answer: 50π/3 ≈ 52.36 cm.

Example 2 — Sign by quadrant. If sin x = 3/5 and x lies in Q-II, find tan x. Step 1 — cos²x = 1 − 9/25 = 16/25 ⇒ cos x = ±4/5; in Q-II cos is negative, so cos x = −4/5. Step 2 — tan x = sin x / cos x = (3/5) / (−4/5) = −3/4. Answer: −3/4.

Example 3 — Sum identity. Find sin 75°. Step 1 — 75° = 45° + 30°. Step 2 — sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (1/√2)(√3/2) + (1/√2)(1/2) = (√3 + 1)/(2√2). Answer: (√3 + 1)/(2√2).

Example 4 — Sum-to-product. Simplify (cos 7x + cos 5x)/(sin 7x − sin 5x). Step 1 — Numerator: cos 7x + cos 5x = 2 cos 6x cos x. Step 2 — Denominator: sin 7x − sin 5x = 2 cos 6x sin x. Step 3 — Ratio = cos x / sin x = cot x. Answer: cot x.

Example 5 — Triple-angle. Find sin 3π/8 given cos(3π/8) value not directly known; instead evaluate cos 3π using the identity cos 3x = 4 cos³x − 3 cos x with x = π. Step 1 — cos π = −1. Step 2 — cos 3π = 4(−1)³ − 3(−1) = −4 + 3 = −1. Step 3 — Confirms cos 3π = cos(π + 2π) = cos π = −1. ✓ Answer: −1.