Sec 3 A-Math Trigonometric Functions & Graphs notes

An angle of any magnitude is measured anticlockwise from the positive x-axis. Its sine is the y-coordinate of the point on the unit circle, its cosine the x-coordinate, and its tangent the ratio y/x. This is why the signs follow the CAST pattern: all positive in the first quadrant, then cos only, tan only, sin only going anticlockwise from the fourth.

Radians are the other required unit: π radians = 180°. Convert degrees to radians by multiplying by π/180, and back by multiplying by 180/π. Key stops on the ruler: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π.

The sine row reads 1, 2, 3 under the radicals (√1/2, √2/2, √3/2); cosine reads the same row backwards. Tangent is their ratio. These combine with the CAST signs for any quadrant: cos 210° = −cos 30° = −√3/2, tan 135° = −1, sin(5π/6) = 1/2.

y = sin x starts at (0°, 0), peaks at 90°, crosses zero at 180°, bottoms at 270° and returns to 360° — one full wave per 360°. y = cos x is the same wave shifted 90° left: it starts at its maximum (0°, 1). Both stay between −1 and 1, and both repeat: sin(x + 360°) = sin x.

y = tan x has no maximum or minimum — it climbs through the origin, then shoots off to +∞ as x approaches 90°. The vertical lines x = 90° + k·180° are asymptotes, and the pattern repeats every 180°: tan(x + 180°) = tan x. This 180° period is why tangent equations in 0° ≤ x ≤ 360° normally have two solutions.

For waves of the form y = a sin(bx) + c or y = a cos(bx) + c (a real, b a positive integer, c an integer):

A negative a flips the wave — y = −2 cos x still has amplitude 2, never −2, because amplitude is a distance. The syllabus also asks for forms like y = a sin(x/b) + c: dividing inside the argument multiplies the period, so y = cos(x/2) has period 360° × 2 = 720°. And y = a tan(bx) keeps the 180°/b period and asymptote spacing, with no maximum or minimum.

Because sin x = 1/2 holds for infinitely many angles, the inverse functions are restricted so they return exactly one value each: sin⁻¹ returns angles from −90° to 90°, cos⁻¹ from 0° to 180°, and tan⁻¹ from −90° to 90° (ends excluded). These single answers are the principal values — the calculator button gives exactly these.

Two traps: first, sin⁻¹ x means the inverse sine, never 1/sin x (that is cosec x). Second, the principal value is usually only the first solution of an equation — finding all solutions in the interval is your job, using the graph symmetries from the previous section.

Recall the reciprocal trio: sec A = 1/cos A, cosec A = 1/sin A, cot A = 1/tan A = cos A/sin A. The quotient identity tan A = sin A/cos A turns mixed sin-and-tan expressions into single-name ones: tan θ cos θ = sin θ. The first Pythagorean identity comes straight from the unit circle (x² + y² = 1); dividing it by cos²A or sin²A gives the other two.

The routine: (1) rearrange so the trigonometric name stands alone, e.g. sin x = k; (2) take the principal value with the calculator; (3) find all solutions in the required interval using the symmetries; (4) round as instructed — 1 decimal place for angles in degrees, 3 significant figures otherwise. In radians, pair α with π − α (sine) or 2π − α (cosine) and α + π (tangent).

Quadratics in disguise appear often. With t = sin x, an equation like 2 sin²x − sin x − 1 = 0 becomes 2t² − t − 1 = 0; solve the quadratic, then run the trig routine for each value of t, remembering −1 ≤ sin x ≤ 1 rules out impossible roots. Equations such as sin 2x = 1/2 need the interval doubled first: for 0° ≤ x ≤ 360°, solve for 2x in 0° ≤ 2x ≤ 720°, then halve every solution.

An identity proof is a one-sided rewrite: pick the more complicated side, convert everything to sines and cosines, apply the Pythagorean and quotient identities, and simplify until it equals the other side. Never treat it as an equation — no "solving for x", no multiplying both sides, and never write equals signs between steps that are not yet justified.