Mathematics 4024 · Syllabus tracker
2D Trigonometry
Pythagoras, sin, cos and tan in right-angled triangles, elevation, depression and bearings, the sine and cosine rules and ½ab sin C.
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2D Trigonometry on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Pythagoras, SOH CAH TOA, elevation, depression and bearings, the sine and cosine rules, and ½ab sin C.
1Pythagoras' theorem
- In a right-angled triangle: a2 + b2 = c2, where c is the hypotenuse (opposite the right angle, the longest side).
- Finding the hypotenuse: square, add, square root. A shorter side: square, subtract, square root.
- Pythagorean triples to recognise: 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and multiples like 6-8-10).
Formulas
Worked example
A ladder 6.5 m long leans against a wall with its foot 2.5 m from the wall. How high up the wall does it reach?
- Hypotenuse 6.5: h2 = 6.52 − 2.52.
- = 42.25 − 6.25 = 36.
6 m
2Sin, cos and tan (SOH CAH TOA)
- Label the sides from the angle: Opposite, Adjacent (next to the angle), Hypotenuse.
- Choose the ratio that uses the two sides involved (one known, one wanted).
- Finding an angle: use the inverse (sin⁻¹, cos⁻¹, tan⁻¹). Calculator in degrees.
| 0° | 30° | 45° | 60° | 90° | |
|---|---|---|---|---|---|
| sin | 0 | ½ | √2/2 | √3/2 | 1 |
| cos | 1 | √3/2 | √2/2 | ½ | 0 |
| tan | 0 | √3/3 | 1 | √3 | — |
Formulas
Worked example
In a right-angled triangle the angle is 52° and the adjacent side is 8 cm. Find the opposite side.
- O and A → tan.
- O = 8 × tan 52° = 10.239…
10.2 cm (3 s.f.)
3Elevation, depression and bearings
- Angle of elevation: measured up from the horizontal. Angle of depression: down from the horizontal.
- The angle of depression from A to B equals the angle of elevation from B to A (alternate angles).
- Bearings questions: draw north lines, mark the bearings, find a right-angled (or other) triangle.
- The shortest distance from a point to a line is the perpendicular distance.
Worked example
From a point 20 m from the foot of a tower, the angle of elevation of the top is 35°. Find the height.
- tan 35° = h ÷ 20.
- h = 20 × tan 35° = 14.0…
14.0 m (3 s.f.)
4Sine rule and cosine rule
- For triangles without a right angle. Label sides a, b, c opposite angles A, B, C.
- Sine rule: use when you know a side and its opposite angle (plus one more).
- Cosine rule: use with two sides and the angle between them (to find the third side) or all three sides (to find an angle).
- A negative cosine means an obtuse angle. sin x = sin(180° − x): the sine rule may give an obtuse answer (the ambiguous case).
Formulas
Worked example
Triangle PQR: PQ = 6 cm, PR = 9 cm, angle P = 50°. Find QR.
- QR2 = 62 + 92 − 2 × 6 × 9 × cos 50°.
- = 117 − 69.42 = 47.58.
QR = 6.90 cm (3 s.f.)
5Area of any triangle
- Area = ½ab sin C, where C is the angle between sides a and b.
- Use it backwards to find an angle or a side when the area is given.
Formulas
Worked example
A triangle has sides 8 cm and 10 cm with 30° between them. Find its area.
- ½ × 8 × 10 × sin 30°.
- = 40 × 0.5.
20 cm²
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Using SOH CAH TOA without a right angle | Use the sine or cosine rule |
| Forgetting to square root at the end of Pythagoras or the cosine rule | c2 = 49 → c = 7 |
| Adding in Pythagoras when finding a shorter side | Shorter side: subtract |
| Calculator in radians | Set it to degrees |
| Giving only the acute angle when the triangle is obtuse | The obtuse angle is 180° − acute |
★In the exam
- Draw and label the triangle, marking O, A, H or a, b, c, A, B, C.
- Keep full calculator values between steps; round only at the end (angles to 1 d.p.).
- Non-calculator paper: use the exact-value table.