Trigonometric ratios of acute angles are generalised here to trigonometric functions of any angle. Angles are measured in degrees or in radians, related by \(\pi\text{ radian}=180^\circ\), and an arc of length \(l\) in a circle of radius \(r\) subtends \(\theta=l/r\) so \(l=r\theta\). Using a unit circle, \(\cos x\) and \(\sin x\) are the coordinates of a point, giving \(\sin^2 x+\cos^2 x=1\), \(1+\tan^2 x=\sec^2 x\) and \(1+\cot^2 x=\operatorname{cosec}^2 x\). Sine and cosine are periodic with period \(2\pi\); their signs vary by quadrant. The chapter derives the sum and difference formulas, e.g. \(\cos(x+y)=\cos x\cos y-\sin x\sin y\) and \(\sin(x+y)=\sin x\cos y+\cos x\sin y\), the double- and triple-angle identities such as \(\sin 2x=2\sin x\cos x\) and \(\cos 2x=1-2\sin^2 x\), and the product-to-sum and sum-to-product transformations. These identities let us evaluate functions at angles like \(15^\circ\) and prove a wide range of trigonometric equalities.
Angles: degree and radian measureArc length \(l=r\theta\)Trigonometric functions on the unit circleSigns, domain and range by quadrantSum and difference identitiesDouble, triple and half-angle formulaeProduct-to-sum and sum-to-product formulae
Key terms
Radian measure
Angle subtended at the centre by an arc equal to the radius; \(\pi\text{ rad}=180^\circ\).
Arc length relation
In a circle of radius \(r\), \(l=r\theta\) where \(\theta\) is in radians.
Pythagorean identity
\(\sin^2 x+\cos^2 x=1\), with \(1+\tan^2 x=\sec^2 x\) and \(1+\cot^2 x=\operatorname{cosec}^2 x\).
