Unit 1: Fundamentals of trigonometry

Trigonometry studies the relationships between the angles and sides of triangles and, more generally, the periodic functions those relationships generate. It begins with the right triangle and extends to the whole coordinate plane through the unit circle.

• Angle: the rotation from an initial ray to a terminal ray about a vertex; positive when anticlockwise, negative when clockwise.
• Two measures: degrees (a full turn = 360°) and radians (a full turn = 2π); 1° = 60′ (minutes), 1′ = 60″ (seconds).
• Standard position: vertex at the origin, initial side along the positive x-axis; the quadrant of the terminal side fixes the signs of ratios.
• Sign convention (ASTC): in quadrants I, II, III, IV the positive ratios are All, Sine, Tangent, Cosine respectively.
• Pythagorean base: for a right triangle, hypotenuse² = perpendicular² + base², the identity every ratio relation rests on.

II. Trigonometric Ratios

The six ratios defined by comparing pairs of sides of a right triangle, then generalised to any angle via coordinates.

A. Trigonometric ratios

For a right triangle with acute angle θ, opposite side (perpendicular) p, adjacent side (base) b, and hypotenuse h.

• Primary three: ratios formed with the hypotenuse.
  
  TEXT

    sin θ = p/h    cos θ = b/h    tan θ = p/b

• Reciprocal three: inverses of the primary ratios.
  
  TEXT

    cosec θ = h/p = 1/sin θ
    sec θ   = h/b = 1/cos θ
    cot θ   = b/p = 1/tan θ

• Quotient links: tangent and cotangent in terms of sine and cosine.
  
  TEXT

    tan θ = sin θ / cos θ     cot θ = cos θ / sin θ

• General angle: with a point P(x, y) on the terminal side at distance r = √(x²+y²), sin θ = y/r, cos θ = x/r, tan θ = y/x. This extends the ratios beyond acute angles and gives them signs by quadrant.
• Worked example: a triangle with p = 3, b = 4 gives h = 5, so sin θ = 3/5 = 0.6 and tan θ = 3/4 = 0.75.

B. Trigonometric ratios of some specific angles

Exact values for the standard angles arise from the 45°–45°–90° and 30°–60°–90° triangles.

• Standard table: angles in degrees (and radians).

  θ	0°	30° (π/6)	45° (π/4)	60° (π/3)	90° (π/2)
  sin	0	1/2	1/√2	√3/2	1
  cos	1	√3/2	1/√2	1/2	0
  tan	0	1/√3	1	√3	∞

• Memory pattern: sin values follow √0/2, √1/2, √2/2, √3/2, √4/2 across 0°–90°; cos is the same sequence reversed.

• Beyond 90°: use reference angles and ASTC, e.g. sin 120° = sin(180°−60°) = √3/2, cos 210° = −cos 30° = −√3/2.

• Undefined points: tan 90° and cot 0° are undefined because their denominators (cos 90°, sin 0°) equal zero.

C. Trigonometric identities

Equations true for every admissible value of the angle, used to simplify expressions and prove relations.

• Pythagorean identities: obtained by dividing p²+b²=h² through by h², b², p².
  
  TEXT

    sin²θ + cos²θ = 1
    1 + tan²θ = sec²θ
    1 + cot²θ = cosec²θ

• Reciprocal identities: sin θ·cosec θ = 1, cos θ·sec θ = 1, tan θ·cot θ = 1.
• Cofunction identities: complementary angles swap function pairs, e.g. sin(90°−θ) = cos θ, tan(90°−θ) = cot θ.
• Worked example: simplify (1 − sin²θ)·sec²θ = cos²θ · (1/cos²θ) = 1, using sin²θ + cos²θ = 1.

III. Trigonometric Functions

Reinterpreting the ratios as functions of a real variable, with the unit circle replacing the right triangle.

A. Trigonometric functions

A function assigns to each angle (or real number) one value of a ratio; the unit circle (r = 1) makes sin and cos the coordinates of a point.

• Unit-circle definition: for the point P(cos θ, sin θ) on the circle x²+y²=1, the x-coordinate is cos θ and the y-coordinate is sin θ.
• Periodicity: sine and cosine repeat every 2π, tangent every π.
  
  TEXT

    sin(θ + 2π) = sin θ    cos(θ + 2π) = cos θ    tan(θ + π) = tan θ

• Even and odd: cosine is even, sine and tangent are odd.
  
  TEXT

    cos(−θ) = cos θ     sin(−θ) = −sin θ     tan(−θ) = −tan θ

• Bounded pair: since coordinates on the unit circle satisfy |x|≤1, |y|≤1, both sin θ and cos θ stay within [−1, 1], while tan θ is unbounded.

B. Relation between radian and real numbers

Radian measure lets an angle be identified with a real number, so trigonometric functions become functions from ℝ to ℝ.

• Definition of radian: the angle subtended at the centre of a circle by an arc equal in length to the radius; θ = arc length / radius = s/r.
• Conversion: a full circle of arc 2πr subtends 2π radians, so 2π rad = 360°.
  
  TEXT

    π radians = 180°
    1 radian = 180/π ≈ 57.2958°
    1° = π/180 radian ≈ 0.01745 radian

• Real-number wrapping: wrapping the real number line around the unit circle maps each real x to a point; the arc length x (in radius units) is the radian measure, so "sin x" reads x as a pure number.
• Consequence: because any real x names an angle, sin, cos and tan are defined for the entire real line (subject to their domains), not just triangle angles.
• Worked example: an arc of length 5 cm on a circle of radius 2 cm subtends θ = 5/2 = 2.5 radians ≈ 143.24°.

C. Domain and range of trigonometric functions

The set of permissible inputs and the resulting set of outputs for each function, dictated by where denominators vanish.

• Sine and cosine: defined everywhere; each maps ℝ onto [−1, 1].
  
  TEXT

    Domain: ℝ            Range: [−1, 1]

• Tangent and secant: excluded where cos x = 0, i.e. x = (2n+1)π/2, n ∈ ℤ.
  
  TEXT

    tan x  Domain: ℝ \ {(2n+1)π/2}    Range: ℝ
    sec x  Domain: ℝ \ {(2n+1)π/2}    Range: (−∞,−1] ∪ [1,∞)

• Cotangent and cosecant: excluded where sin x = 0, i.e. x = nπ, n ∈ ℤ.
  
  TEXT

    cot x    Domain: ℝ \ {nπ}    Range: ℝ
    cosec x  Domain: ℝ \ {nπ}    Range: (−∞,−1] ∪ [1,∞)

• Why the gaps: sec and cosec inherit values ≥1 or ≤−1 because they are reciprocals of quantities bounded by 1; they can never land in (−1, 1).

IV. Trigonometric Functions of Sum and Difference of Two Angles

Formulae expressing a ratio of A±B through the ratios of A and B, the engine for expansions, multiple-angle and product formulae.

A. Trigonometric functions of sum and difference of two angles

These compound-angle formulae let any combined angle be broken into known parts; they are proved from the unit-circle distance formula.

• Cosine formulae: note the sign reversal relative to the operation.
  
  TEXT

    cos(A + B) = cos A cos B − sin A sin B
    cos(A − B) = cos A cos B + sin A sin B

• Sine formulae:
  
  TEXT

    sin(A + B) = sin A cos B + cos A sin B
    sin(A − B) = sin A cos B − cos A sin B

• Tangent formulae: valid where all tangents and the denominator are defined.
  
  TEXT

    tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
    tan(A − B) = (tan A − tan B) / (1 + tan A tan B)

• Special cases derived from them:
  • Double angle: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1.
  • Complementary check: cos(90° − B) = cos 90° cos B + sin 90° sin B = sin B, recovering the cofunction identity.
• Worked example: cos 75° = cos(45° + 30°) = cos45°cos30° − sin45°sin30° = (1/√2)(√3/2) − (1/√2)(1/2) = (√3 − 1)/(2√2).

B. Applications and limitations

The sum and difference formulae underpin the wider algebra of trigonometry but carry domain restrictions.

• Product and sum conversions: adding the sine formulae gives 2 sin A cos B = sin(A+B) + sin(A−B), the basis of product-to-sum rules used in integration and wave analysis.
• Restriction on tangent form: tan(A+B) fails when 1 − tan A tan B = 0 (that is, when A+B = 90°), because the tangent of a right angle is undefined; the sine and cosine forms remain valid there.
• Angle synthesis: any multiple of 15° can be built from 30°, 45° and 60°, so exact values such as cos 15° and sin 105° follow directly from these formulae without new triangles.