Quick revision notes · Mathematics 4024 · Chapter 8 of 35

2D Trigonometry

Pythagoras, SOH CAH TOA, elevation, depression and bearings, the sine and cosine rules, and ½ab sin C.

5 sectionsWorked examples Common mistakes

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Pythagoras, SOH CAH TOA, elevation, depression and bearings, the sine and cosine rules, and ½ab sin C.

Pythagoras' theorem
Sin, cos and tan (SOH CAH TOA)
Elevation, depression and bearings
Sine rule and cosine rule
Area of any triangle

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

a² + b² = c²c = hypotenuse

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

sin θ = O/H

cos θ = A/H

tan θ = O/A

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

a/sin A = b/sin B = c/sin Csine rule

a² = b² + c² − 2bc cos Acosine rule (side)

cos A = (b² + c² − a²) / 2bccosine rule (angle)

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

Area = ½ab sin C

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.