A-Level Maths: Radians and Trigonometric Equations

Solving a trigonometric equation over a given interval is one of the most frequent tasks in A level Mathematics papers, and it depends on two things: fluent use of radians and a reliable method for finding every solution, and no extra ones. This material trains both, based on the trigonometry content of the DfE A level Mathematics subject content.

It starts with radian measure. You use the formulae $s=r\theta$ for arc length and $\frac{1}{2}{r}^2\theta$ for the area of a sector, find the area of a segment as a sector minus a triangle, and work backwards from an area to an angle and a perimeter. You then meet the inverse functions $\arcsin$, $\arccos$ and $\arctan$: why sine, cosine and tangent must be restricted to a domain where they are one-one, what the resulting domains and ranges are, and why $\arccos (-\frac{1}{2})$ is $\frac{2\pi }{3}$ and not any other angle with the same cosine.

The equation questions follow a clear progression. Multiple-angle equations such as $\sin 2x=\frac{\sqrt{3}}{2}$ or $\tan (x-{30}^{\circ })=1$ are solved by substituting $u$ for the compound angle and transforming the interval. Quadratic equations in $\cos x$ or $\tan x$ are factorised, with impossible values such as $\cos x=-2$ rejected. Identities turn mixed equations into single-function ones: ${\sin }^{2}x+{\cos }^{2}x=1$, ${\sec }^{2}x=1+{\tan }^{2}x$, the double angle formulae, and the form $R\sin (x+\alpha )$. A recurring point is never to divide by $\sin x$ or $\cos x$, because solutions are lost; you factorise instead. Answers are given in degrees or radians as the question asks.

The material offers three formats. The quiz has 12 multiple-choice questions with full explanations that show the interval changes and the typical errors behind the wrong options, such as missing the second solution or keeping values outside the interval. The flashcards collect the radian formulae, the domains and ranges of the inverse functions and the rules for generating further solutions. The written work gives 8 longer questions to answer by hand, including an equation in $R\sin (x-\alpha )$ form, an equation that needs the double angle formula for $\cos 2x$ and a short proof followed by an equation.

• Use radian measure for arc length, sector area and segment area
• State the domains and ranges of arcsin, arccos and arctan and evaluate them exactly
• Solve trigonometric equations with multiple or compound angles in a given interval
• Solve quadratic equations in a trigonometric function
• Use Pythagorean identities, double angle formulae and R sin(x + α) to solve equations
• Find all solutions in degrees or radians without losing or adding any

Practice material written by Zestly, based on the DfE A level Mathematics subject content (trigonometry: radian measure, arc length and sector area, inverse trigonometric functions, solving trigonometric equations in a given interval, identities).