Know the exact values, the shapes and symmetries of the three graphs, and the two identities \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) and \(\sin^2\theta + \cos^2\theta = 1\). Most equation questions ask 'how many solutions' - a sketch of the graph in the given interval is faster and safer than listing them all.
Contents: Exact values and graphs (MM4.3, MM4.4)Identities and equations (MM4.5, MM4.6)Must-knowTrapsReal questions
Exact values and graphs (MM4.3, MM4.4)
- \(\sin\): \(0, \frac12, \frac{\sqrt2}{2}, \frac{\sqrt3}{2}, 1\) for \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\); \(\cos\) runs the same list backwards; \(\tan\): \(0, \frac{1}{\sqrt3}, 1, \sqrt3\), undefined.
- \(\sin\) and \(\cos\) have period \(360^\circ\) (\(2\pi\)), range \([-1, 1]\); \(\tan\) has period \(180^\circ\) (\(\pi\)) and asymptotes at \(90^\circ + 180^\circ k\).
- \(\cos\) is even (\(\cos(-x) = \cos x\)), \(\sin\) and \(\tan\) are odd. \(\cos x = \sin(x + 90^\circ)\). \(\sin(180^\circ - x) = \sin x\), \(\cos(360^\circ - x) = \cos x\), \(\tan(x + 180^\circ) = \tan x\).
- \(y = a\sin(bx) + c\) has amplitude \(|a|\), period \(\frac{360^\circ}{b}\) and midline \(y = c\).
Identities and equations (MM4.5, MM4.6)
- \(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\); \(\sin^2\theta + \cos^2\theta = 1\) (so \(\cos^2\theta = 1 - \sin^2\theta\) turns \(2\cos^2x + 3\sin x - 3 = 0\) into a quadratic in \(\sin x\)).
- Solving \(\sin(kx + a) = c\): find the solutions for \(u = kx + a\) over the stretched interval, then convert. The number of solutions is usually \(k\) times what it would be for \(\sin u = c\).
- Sign check with CAST or the graph: \(\sin x = \frac12\) has \(30^\circ\) and \(150^\circ\); \(\cos x = -\frac12\) has \(120^\circ\) and \(240^\circ\); \(\tan x = -1\) has \(135^\circ\) and \(315^\circ\).
- \(\sin x = \sin\alpha\) does not imply \(x = \alpha\); it gives \(x = \alpha\) or \(180^\circ - \alpha\) (plus multiples of \(360^\circ\)).
- Never divide by \(\cos x\) or \(\sin x\): factorise instead, or you lose the solutions where it is zero.
Must-know (no formula booklet)
- \(\sin 30^\circ = \frac12\), \(\cos 30^\circ = \frac{\sqrt3}{2}\), \(\tan 60^\circ = \sqrt3\)
- \(\sin^2 x + \cos^2 x = 1\)
- period of \(\sin(kx)\) is \(360^\circ/k\)
- \(\sin(180^\circ - x) = \sin x\)
Classic traps
- Check the interval's endpoints: \(0 < x < 360^\circ\) excludes \(0^\circ\) and \(360^\circ\) but includes \(180^\circ\).
- \(\sin^2 x = \frac34\) has four solutions in a full turn, not two.
- Radians vs degrees: \(\sin x = \frac{x}{10}\) only makes sense in radians.
Worked example
Q. How many solutions has \(\sin 2x = \frac12\) for \(0 \leq x < 2\pi\)?
A. \(2x\) runs over \([0, 4\pi)\) - two periods, two solutions each: four.