Cambridge International AS & A Level Mathematics 9709 (2026-2027) · Paper 3: Pure Mathematics 3 · official syllabus
TRIGONOMETRY LAB
Learn it by playing it
Four short missions on flipping sec into cos, trading tan^2 for sec^2 with the Pythagorean identity, building an exact sin 75 deg from two standard angles, and merging 3 cos x + 4 sin x into a single wave. When a mission glows green, read why it worked, then take the next one.
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Mission 1 of 4
Meet the reciprocals
sec x = 2.5 and you need cos x - no new mathematics, just a flip. The dial shows the value of sec x for the cos x you set. Slide cos x until the dial reads exactly 2.5.
Before you slide: sec x is defined as...
Your dial reads sec x = 1.43 with cos x = 0.70 - the target is 2.5, so cos x = 1 / 2.5.
Study notes
Played the lab above? These notes are the reference version of what you just felt - use them to check exact formulas and the exam wording.
The reciprocal functions: secant, cosecant, cotangent 3.3
sec x = 1 / cos x, cosec x = 1 / sin x, cot x = 1 / tan x = cos x / sin x. The pairings matter: sec goes with cos, cosec goes with sin.
Flipping is the first move: sec x = 2.5 means cos x = 1 / 2.5 = 0.4 exactly, and cosec x = 2 means sin x = 1/2.
Their graphs inherit asymptotes where the parent function is zero: sec x blows up at x = 90 deg (pi/2) because cos x = 0 there, and cosec x blows up at x = 0 and 180 deg.
Since |cos x| <= 1, sec x can never sit between -1 and 1 - the same range fence applies to cosec x.
The Pythagorean identities with sec, cosec and cot 3.3
Divide sin^2 x + cos^2 x = 1 by cos^2 x to get tan^2 x + 1 = sec^2 x; divide by sin^2 x to get 1 + cot^2 x = cosec^2 x. Two identities, one parent.
They trade one function for another without any angle: tan x = 2 gives sec^2 x = 5, so sec x = sqrt(5) = 2.236 (rounded to 3 d.p.) for an acute angle.
Equations mixing the families become quadratics: 3 tan^2 x + 2 sec x - 5 = 0 turns into 3(sec^2 x - 1) + 2 sec x - 5 = 0, a quadratic in sec x. Solve, then flip back to cos x.
Choose the sign from the quadrant: sec^2 x = 5 gives sec x = +sqrt(5) when x is acute, but -sqrt(5) in the third quadrant where cos x is negative.
Compound-angle formulae 3.3
sin(A + B) = sin A cos B + cos A sin B and sin(A - B) = sin A cos B - cos A sin B: sine keeps the sign and mixes the functions.
cos(A + B) = cos A cos B - sin A sin B and cos(A - B) = cos A cos B + sin A sin B: cosine flips the sign and keeps the functions paired.
tan(A + B) = (tan A + tan B) / (1 - tan A tan B), from dividing the sine and cosine versions by cos A cos B.
Exact-value questions build unusual angles from 30, 45 and 60: sin 75 deg = sin(45 + 30) = (sqrt(6) + sqrt(2))/4 = 0.9659 (rounded to 4 d.p.). The surd form is the answer; the decimal is the check.
Double-angle formulae 3.3
Set A = B in the compound formulae: sin 2A = 2 sin A cos A, and tan 2A = 2 tan A / (1 - tan^2 A). Worked example: tan A = 2 gives tan 2A = 4 / (1 - 4) = -4/3.
cos 2A comes in three shapes: cos^2 A - sin^2 A = 2 cos^2 A - 1 = 1 - 2 sin^2 A. Pick the one that matches the rest of the equation.
The rearrangements sin^2 A = (1 - cos 2A)/2 and cos^2 A = (1 + cos 2A)/2 remove powers - they are how later integration of sin^2 x becomes possible, and how equations in sin^2 x become linear in cos 2x.
Equations like sin 2x = sin x become 2 sin x cos x = sin x: never divide by sin x (that loses the sin x = 0 solutions) - factor instead.
The harmonic form R cos(x - alpha) 3.3
a cos x + b sin x is one wave: match R cos(x - a) = R cos x cos a + R sin x sin a, so R cos a = a and R sin a = b, giving R = sqrt(a^2 + b^2) and tan a = b / a.
Worked example: 3 cos x + 4 sin x = 5 cos(x - 53.13 deg), with 53.13 rounded to 2 d.p., because R = sqrt(9 + 16) = 5 and tan a = 4 / 3.
The form hands you the max and min for free: the maximum of a cos x + b sin x is R, the minimum is -R, and a cos x + b sin x = k has solutions only when |k| <= R.
R sin(x + a) = R sin x cos a + R cos x sin a fits a cos x + b sin x with the roles swapped; radians work identically - only the units of a change.
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