Geometry

Explanation

• Standard areas include rectangle $lw$, triangle $\frac12bh$, parallelogram $bh$, trapezium $\frac12(a+b)h$ and circle $\pi r^2$; circumference is $2\pi r$. Prism volume is cross-sectional area times length; cylinder volume is $\pi r^2h$, curved area $2\pi rh$ and total area $2\pi rh+2\pi r^2$.
• Sphere volume and area are $\frac43\pi r^3$ and $4\pi r^2$; cone volume and total area are $\frac13\pi r^2h$ and $\pi rl+\pi r^2$; pyramid volume is $\frac13\times\text{base area}\times\text{perpendicular height}$. Prism and pyramid surface areas sum all exposed faces.
• Angle properties cover parallel and intersecting lines, triangles, all special quadrilaterals and polygons.
• Circle facts include centre angle twice circumference angle, same-segment angles, semicircle right angle, opposite cyclic angles, radius–tangent perpendicularity, alternate segment, centre-perpendicular bisecting a chord, and equal tangents from an external point.
• Examiners expect compatible units and named reasons.

Explanation

• A geometrical proof is a connected chain of statements, each justified by a definition, theorem or established result. Precise labels such as $\angle ABC$ identify the vertex and arms unambiguously.
• Congruence can prove corresponding sides or angles equal; valid tests include SSS, SAS, ASA and RHS. Similarity, parallel-line facts and circle theorems may also supply steps.
• A diagram must not be assumed to scale, and the desired conclusion cannot be used as a premise.
• For circle proofs, the relevant chord, radius, tangent or arc should be identified.
• Examiners expect reasons beside key statements and a final sentence that explicitly establishes the proposition.

Explanation

• Right-angled triangles use $\sin=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos=\frac{\text{adjacent}}{\text{hypotenuse}}$ and $\tan=\frac{\text{opposite}}{\text{adjacent}}$; inverse ratios recover an unknown acute angle from side lengths. For a scalene triangle, the sine rule $\frac a{\sin A}=\frac b{\sin B}=\frac c{\sin C}$ needs an opposite side–angle pair.
• The cosine rule $c^2=a^2+b^2-2ab\cos C$ uses three sides or two sides and their included angle.
• Area is $\frac12ab\sin C$.
• A sine-rule angle may have an alternative because $\sin\theta=\sin(180^\circ-\theta)$; the triangle sum decides validity.
• Examiners expect correctly paired labels, unrounded working and final rounding only as requested.

Explanation

• In a right-angled triangle, $a^2+b^2=c^2$, where $c$ is the hypotenuse opposite the right angle. The theorem can find a missing side or verify a right angle.
• Required familiar triples include $3,4,5$, $5,12,13$, $8,15,17$ and $7,24,25$, together with their multiples.
• In three dimensions, a face diagonal can be found first and used in a second right-angled triangle; for a cuboid this combines to $d^2=l^2+w^2+h^2$.
• Pythagoras is valid only when the right angle is established, and a calculated length must be positive.
• Examiners expect the longest side to be treated as the hypotenuse and each 3D right-angled cross-section to be identified.

Explanation

• A 3D trigonometry problem should be reduced to a labelled right-angled cross-section.
• The angle between a line and a plane is the angle between the line and its perpendicular projection onto that plane, not an angle with an arbitrary edge.
• For the angle between two planes, take a cross-section perpendicular to their line of intersection; the angle between the two cross-section lines is the dihedral angle.
• Pythagoras often supplies a face diagonal or projection before sine, cosine or tangent is applied.
• Examiners expect the relevant projection, right angle and required angle to be identified, because using a visually convenient but incorrect triangle gives the wrong ratio.

Explanation

• In degrees, $y=\sin x$ and $y=\cos x$ have period $360^\circ$ and range $-1\leq y\leq1$.
• Sine passes through $0$ at multiples of $180^\circ$; cosine begins at $1$ when $x=0^\circ$.
• The tangent graph has period $180^\circ$, zeros at multiples of $180^\circ$ and vertical asymptotes at $90^\circ+180^\circ n$.
• Graphs extend to angles of any size by repeating these periods in both directions.
• Examiners expect smooth curves through exact key points, separate tangent branches that never cross their asymptotes, and correct use of graphs to read signs, intersections or approximate solutions.

Explanation

• For any positive angle up to $360^\circ$, measured anticlockwise from the positive $x$-axis, a point $(x,y)$ at distance $r$ from the origin gives $\cos\theta=\frac xr$, $\sin\theta=\frac yr$ and $\tan\theta=\frac yx$.
• Sine is positive in quadrants I and II, cosine in I and IV, and tangent in I and III.
• A reference angle gives the magnitude, while the quadrant fixes the sign and full angle.
• Inverse trigonometric functions often return only a principal value, so quadrant information must be applied.
• Examiners expect degree mode and an angle within the stated quadrant or interval.

Explanation

• A $30^\circ$–$60^\circ$–$90^\circ$ triangle has side ratio $1:\sqrt3:2$, opposite $30^\circ$, $60^\circ$ and $90^\circ$ respectively. A $45^\circ$–$45^\circ$–$90^\circ$ triangle has ratio $1:1:\sqrt2$.
• These ratios follow by halving an equilateral triangle or bisecting a square, and they generate exact trigonometric values such as $\sin30^\circ=\frac12$, $\cos30^\circ=\frac{\sqrt3}{2}$ and $\sin45^\circ=\frac{\sqrt2}{2}$.
• They also produce exact lengths in regular polygons and composite geometry.
• Scale every side by the same factor and retain exact surds unless a decimal is requested.
• Examiners expect each ratio length to be matched to the angle opposite it.

Explanation

• The identities $\tan x=\frac{\sin x}{\cos x}$ and $\sin^2x+\cos^2x=1$ allow one trigonometric form to be replaced by another. Useful rearrangements are $1-\sin^2x=\cos^2x$ and $1-\cos^2x=\sin^2x$.
• They also solve equations by replacing squared terms or tangent with sine and cosine before factorising. When finding a missing value, the quadrant determines the sign of a square root.
• In an identity proof, start from one side and transform it through valid equal expressions until the other is reached.
• A common denominator often exposes an identity.
• Examiners expect restrictions to be respected whenever a cancelled denominator could be zero.

Explanation

• To solve a simple trigonometric equation, first isolate the trigonometric function and find a reference angle. Use its sign to choose the correct quadrants, then list every solution in the stated interval.
• Sine and cosine repeat every $360^\circ$; tangent repeats every $180^\circ$.
• A squared equation may produce positive and negative function values, so factor or take both square-root possibilities.
• Interval endpoints must be included or excluded exactly as stated, and angles are measured in degrees for this specification.
• Examiners expect all solutions, normally in ascending order, with substitution checks when factorisation or cancellation could introduce or remove a value.