The unit opens with trigonometry in the real world: angles of elevation and depression, then three-figure bearings, then right-angled triangles hidden inside solids. Students choose the ratio, decide whether to multiply or divide, and combine two triangles when one is not enough.
Then the angles 30°, 45° and 60° get exact values from two special triangles, with no calculator, and a sector becomes a fraction of a circle with an arc, an area and a perimeter. The last three lessons prove the circle theorems: chords and tangents, the angle at the centre, and cyclic quadrilaterals with the alternate segment, each from isosceles triangles and angle sums.
A graphic display calculator (GDC) is optional all the way through. Lessons 1, 2, 3 and 5 have a closed “With a GDC” section to open if you have one; lessons 4, 6, 7 and 8 are done by hand.
The lessons, in order
- Lesson 1 · Elevation, depression and choosing a ratio Angle of elevation and angle of depression, choosing sine, cosine or tangent, multiplying or dividing, and two-triangle problems. Move a lookout's height and sight angle and watch the angle at the boat, then pick which side is known and which is wanted and see the ratio. GDC optional
- Lesson 2 · Three-figure bearings Three-figure bearings, back bearings, east, west, north and south components, and bearing problems with right-angled triangles. Move a ship's bearing and distance and watch its east and north parts and the back bearing, then place a point and read its bearing. GDC optional
- Lesson 3 · Right-angled triangles in 3D The space diagonal of a cuboid by Pythagoras twice, sloping edges of pyramids, and the angle between a line and a plane. Change the three edges of a box and watch the floor diagonal and the space diagonal, then change a pyramid's base and height and read the angle of an edge. GDC optional
- Lesson 4 · Exact values for 30°, 45° and 60° Exact values of sine, cosine and tangent for 30°, 45° and 60° from half a square and half an equilateral triangle, used in problems with surds. Slide the size of a right isosceles triangle and of half an equilateral triangle and watch the ratios stay exact. No calculator
- Lesson 5 · Arcs and sectors Arc length and sector area as a fraction of 360°, the perimeter of a sector, and finding an angle or radius from an arc or an area. Change a sector's angle and radius and watch the fraction, the arc, the area and the perimeter. GDC optional
- Lesson 6 · Chords, tangents and radii Chord and radius symmetry, the tangent perpendicular to the radius, equal tangents from a point, each proved, and used with Pythagoras. Slide a chord away from the centre until it becomes a tangent, then move a point away from a circle and watch the two tangent lengths. No calculator
- Lesson 7 · The angle at the centre and angles in a segment The angle at the centre is twice the angle at the circumference, angles in the same segment are equal, the angle in a semicircle is 90°, each proved and then applied with algebra. Fix two points on a circle, move a third round the major arc, and then watch the proof built from two isosceles triangles. No calculator
- Lesson 8 · Cyclic quadrilaterals and the alternate segment Opposite angles of a cyclic quadrilateral, the exterior angle, the alternate segment theorem, each proved, and mixed problems with algebra. Slide two arcs of a circle and watch every angle of the inscribed quadrilateral, then move a point round the alternate segment. No calculator
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.