Trigonometry

After calculus, trigonometry is the heaviest-tested topic in 0606, and the one where students most often feel lost while actually being one routine away from competence. The topic splits cleanly: values and graphs (knowledge), identities and equations (method). The exam techniques for the method half have their own technique guide; these notes build the foundations.

Exact values and the unit circle

The exact values of $\sin$, $\cos$, $\tan$ at $0°$, $30°$, $45°$, $60°$, $90°$ ($\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$) must be instant, Paper 1 assumes them. Two derivation anchors if memory blanks: the $1$-$1$-$\sqrt{2}$ right isosceles triangle ($45°$) and the $1$-$\sqrt{3}$-$2$ half-equilateral ($30°$/$60°$).

The unit circle extends them to all angles: a point at angle $\theta$ on the circle of radius $1$ has coordinates $(\cos\theta, \sin\theta)$. From that one picture: which functions are positive in which quadrant (the CAST pattern), the symmetries $\sin(180° - \theta) = \sin\theta$ and $\cos(360° - \theta) = \cos\theta$, and why equations have multiple solutions. Students who learn the circle rather than the CAST mnemonic alone can reconstruct under pressure instead of recalling.

Graphs of sin, cos and tan

$y = a\sin(bx) + c$: amplitude $|a|$, period $360°/b$ ($2\pi/b$ in radians), centre line $y = c$. Cosine is sine shifted left $90°$; tangent has period $180°$, no amplitude, and asymptotes at $90° + 180°n$. Exam sketches award B marks for labelled maxima/minima, intercepts and (for $\tan$) asymptotes, the sketch command is about features, not artistry. Counting intersections with a line (e.g. “state the number of solutions of $\sin 2x = 0.3$ for $0 \le x \le 360°$”) is a pure graph-reading question: sketch, draw the horizontal line, count.

The identities

The working set: $\tan\theta = \sin\theta/\cos\theta$; $\sin^2\theta + \cos^2\theta = 1$ with its rearrangements; dividing through by $\cos^2\theta$ or $\sin^2\theta$ gives $1 + \tan^2\theta = \sec^2\theta$ and $1 + \cot^2\theta = \csc^2\theta$; and the reciprocal trio $\sec = 1/\cos$, $\csc = 1/\sin$, $\cot = 1/\tan$. All memorised, none given. Identity proofs, start from the messier side, convert to $\sin$/$\cos$ when stuck, never cross the equals sign, are drilled in the technique guide.

The R-formula

$a\sin\theta \pm b\cos\theta$ collapses to a single wave $R\sin(\theta \pm \alpha)$ with $R = \sqrt{a^2 + b^2}$, $\tan\alpha = b/a$:

$3\sin\theta + 4\cos\theta = 5\sin(\theta + 53.1°)$

Why it matters: a sum of two waves becomes one function whose maximum is $R$ (minimum $-R$), and whose equations solve by the standard single-function routine. The three-part exam pattern, express, solve, state max/min, is among the most predictable in 0606.

Solving trig equations: the discipline

Range and units first; rearrange to one function; reference angle; all solutions in range via the unit circle; compound arguments ($2x - 30°$) solved over the expanded range before converting back. Never divide by a trig factor that could be zero, factorise. The complete routine with traps is in the technique guide, and missing-solution errors head the common mistakes list.

Worked exam-style question

Solve $2\cos^2 x + \sin x = 1$ for $0° \le x \le 360°$.

Replace $\cos^2 x = 1 - \sin^2 x$: $2(1 - \sin^2 x) + \sin x = 1$ (M, identity used) $\to 2\sin^2 x - \sin x - 1 = 0$ (collected, quadratic in $\sin x$) $\to (2\sin x + 1)(\sin x - 1) = 0$ (M, factorised) $\to \sin x = -\frac{1}{2}$ or $\sin x = 1$ $\sin x = -\frac{1}{2}$: reference $30°$, third/fourth quadrants $\to x = 210°, 330°$ (A, A) $\sin x = 1$: $x = 90°$ (A) $x = 90°, 210°, 330°$

The identity substitution line and the all-quadrants sweep are where scripts diverge, five marks for the systematic, two for the hopeful.

Common mistakes in this topic

• Exact values shaky, Paper 1 bleeding
• Equations solved in the wrong angular unit for the range given (radians)
• Second-quadrant (or third/fourth) solutions missing
• $\sin^2\theta + \cos^2\theta = 1$ known but its rearrangements unrecognised inside expressions
• R-formula $\alpha$ placed in the wrong quadrant

Trig connects forward into calculus (trig derivatives and integrals, radians required) and shares the $\frac{1}{2}ab\sin C$ bridge with circular measure. Weeks 2-3 of the revision plan are built around it.

If trig is the wall between you and your target grade, it’s the most coachable wall in the syllabus, free 1-hour trial class with Teacher Rig, booked on WhatsApp.