Sec 3 E-Math Trigonometry notes

In a right-angled triangle with hypotenuse c (the side opposite the right angle) and legs a and b, a² + b² = c². The theorem computes the third side whenever two are known: legs 6 and 8 give hypotenuse √(36 + 64) = 10. Useful exact triples: 3-4-5, 5-12-13, 8-15-17 (and their multiples).

1. 1Find the hypotenuse

   Legs 9 and 12: c² = 81 + 144 = 225, so c = 15.

2. 2Find a leg

   Hypotenuse 13, one leg 5: b² = 169 − 25 = 144, so b = 12.

3. 3Keep the sides labelled

   The hypotenuse is always the longest side and sits opposite the right angle.

In a right triangle, each acute angle owns three ratios of the sides. Relative to the angle θ: the opposite side faces θ, the adjacent side touches θ (and is not the hypotenuse), and the hypotenuse is opposite the right angle. Then: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj — the memory line SOH CAH TOA. Label the triangle before touching the calculator; most errors are labelling errors.

Sine and cosine extend to obtuse angles by symmetry about 90°: sin(180° − θ) = sin θ and cos(180° − θ) = −cos θ for acute θ. So sin 130° = sin 50° ≈ 0.766, while cos 110° = −cos 70° ≈ −0.342. Sine stays positive across the quadrant; cosine turns negative — the sign that later powers the cosine rule’s ± behaviour.

• sin 150° = sin 30° = 0.5.
• cos 120° = −cos 60° = −0.5.
• The calculator handles this natively: type sin(130) directly and check it agrees with sin(50).

The area of any triangle with sides a, b enclosing angle C is ½ab sin C: half of 8 by 5 with C = 40° gives ½ × 8 × 5 × sin 40° = 12.9 cm² (3 s.f.). The sine rule pairs each side with its opposite angle, a/sin A = b/sin B = c/sin C, and works best when you know a matching side–angle pair. The cosine rule comes in two costumes: c² = a² + b² − 2ab cos C finds a side from two sides and the included angle, and cos C = (a² + b² − c²)/(2ab) finds an angle from three sides.

1. 1Sine rule

   A = 52°, a = 8, B = 70°: b = 8 sin 70°/sin 52° = 9.54.

2. 2Cosine rule (side)

   a = 6, b = 9, C = 62°: c² = 36 + 81 − 2(6)(9)cos 62° = 66.3, so c = 8.14.

3. 3Cosine rule (angle)

   Sides 8, 11, 13: cos(angle opposite 13) = (64 + 121 − 169)/(2 × 8 × 11) = 0.0909, so the angle = 84.8°.

Decision guide: two angles plus any side, or two sides plus a non-included angle → sine rule; two sides plus the included angle, or three sides → cosine rule; two sides plus their included angle for an area → ½ab sin C.

The angle of elevation is measured up from the horizontal (looking up at a tower); the angle of depression is measured down from the horizontal (looking down from a cliff). The two horizontals are parallel, so by alternate angles the depression from above equals the elevation from below. Bearings are compass directions measured clockwise from north and written with three digits: 048°, 125°, 308°.

1. 1Elevation to distance

   A 45 m tower shows elevation 32° from a point: distance = 45/tan 32° = 72.0 m.

2. 2Depression to distance

   From an 80 m cliff a boat shows depression 25°: horizontal distance = 80/tan 25° = 172 m (3 s.f.).

3. 3Draw before computing

   Sketch the right triangle, mark the horizontal and the given angle, then choose tan (two legs) or sin/cos (hypotenuse involved).

Three-dimensional problems stack two right triangles: for example, in a cuboid the space diagonal needs Pythagoras twice (base diagonal, then the 3D diagonal), and the angle it makes with the base uses tan between the height and the base diagonal. Draw the base triangle and the vertical triangle separately.