This note covers Pythagoras' theorem, right-angled trigonometry, angles of elevation and depression, and bearings. At Higher tier it also covers exact trigonometric values, the sine and cosine rules, the area formula \(\tfrac12 ab\sin C\), and problems in three dimensions. These tools let you find lengths and angles that cannot be measured directly, such as the height of a tower or the distance across a lake.
Key ideas
Pythagoras' theorem. In a right-angled triangle, the square on the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares on the other two sides: \[ a^2 + b^2 = c^2 \] To find the hypotenuse, add the squares and take the square root. To find a shorter side, subtract. The converse also holds: if three sides satisfy \(a^2 + b^2 = c^2\), the triangle is right-angled. Whole-number examples such as 3, 4, 5 and 5, 12, 13 are called Pythagorean triples.
Right-angled trigonometry. Label the sides relative to the angle \(\theta\): the hypotenuse is opposite the right angle, the opposite side is across from \(\theta\), and the adjacent side is next to \(\theta\). \[ \sin\theta = \frac{\text{opp}}{\text{hyp}}, \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \qquad \tan\theta = \frac{\text{opp}}{\text{adj}} \]
- To find a side, choose the ratio that contains the known side and the wanted side, then rearrange.
- To find an angle, use the inverse function on your calculator, such as \(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\).
- Check that your calculator is in degree mode.
Angles of elevation and depression. The angle of elevation is measured upwards from the horizontal. The angle of depression is measured downwards from the horizontal. The angle of depression from \(A\) to \(B\) equals the angle of elevation from \(B\) to \(A\) (alternate angles).
Bearings. A bearing is an angle measured clockwise from north, written with three figures, for example \(045^\circ\) or \(260^\circ\). The back bearing differs by \(180^\circ\).
Exact values (Higher). You should know these without a calculator:
- \(\sin 30^\circ = \tfrac12\), \(\cos 30^\circ = \tfrac{\sqrt3}{2}\), \(\tan 30^\circ = \tfrac{1}{\sqrt3}\).
- \(\sin 45^\circ = \cos 45^\circ = \tfrac{\sqrt2}{2}\), \(\tan 45^\circ = 1\).
- \(\sin 60^\circ = \tfrac{\sqrt3}{2}\), \(\cos 60^\circ = \tfrac12\), \(\tan 60^\circ = \sqrt3\).
- \(\sin 0^\circ = 0\), \(\cos 0^\circ = 1\), \(\sin 90^\circ = 1\), \(\cos 90^\circ = 0\).
Non-right-angled triangles (Higher). Label the triangle with sides \(a, b, c\) opposite angles \(A, B, C\).
- Sine rule: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\). Use it when you know an angle and its opposite side, plus one more piece of information.
- Cosine rule: \(a^2 = b^2 + c^2 - 2bc\cos A\). Use it when you know two sides and the angle between them, or all three sides. To find an angle, rearrange to \(\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}\).
- Area: \(\text{area} = \tfrac12 ab\sin C\), where \(C\) is the angle between sides \(a\) and \(b\).
Three-dimensional problems (Higher). Find a right-angled triangle inside the solid and apply Pythagoras or trigonometry. For a cuboid with edges \(l, w, h\), the space diagonal is \(\sqrt{l^2 + w^2 + h^2}\). The angle between a line and a plane is the angle between the line and its shadow (projection) on the plane.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| \(a^2 + b^2 = c^2\) | Pythagoras: sides of a right-angled triangle, with \(c\) the hypotenuse. |
| \(\sin\theta = \dfrac{\text{opp}}{\text{hyp}}\) | Right-angled triangle with the opposite side and hypotenuse involved. |
| \(\cos\theta = \dfrac{\text{adj}}{\text{hyp}}\) | Adjacent side and hypotenuse involved. |
| \(\tan\theta = \dfrac{\text{opp}}{\text{adj}}\) | Opposite and adjacent sides involved. |
| \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\) | Sine rule (Higher), for a matching side and angle pair. |
| \(a^2 = b^2 + c^2 - 2bc\cos A\) | Cosine rule (Higher), for two sides and the included angle, or for three sides. |
| \(\tfrac12 ab\sin C\) | Area of any triangle (Higher). |
Worked examples
Example 1 A ladder 5.2 m long leans against a vertical wall. Its foot is 1.5 m from the wall on level ground. Find (a) how high up the wall the ladder reaches, (b) the angle the ladder makes with the ground.
(a) The ladder is the hypotenuse. Let the height be \(h\).
\(h^2 = 5.2^2 - 1.5^2 = 27.04 - 2.25 = 24.79\), so \(h = \sqrt{24.79} = 4.98\) m (3 s.f.).
(b) The side 1.5 m is adjacent to the angle \(\theta\) at the ground, and 5.2 m is the hypotenuse, so use cosine.
\(\cos\theta = \dfrac{1.5}{5.2} = 0.2885\). Use the inverse cosine function: \(\theta = 73.2^\circ\) (3 s.f.).
Answer: (a) 4.98 m (b) \(73.2^\circ\)
Example 2 A student stands 45 m from the foot of a tower on level ground. Her eyes are 1.6 m above the ground, and the angle of elevation of the top of the tower is \(38^\circ\). Find the height of the tower.
Draw a horizontal line from her eyes to the tower. The opposite side to \(38^\circ\) is the height above eye level, and the adjacent side is 45 m. Use tangent.
\(\tan 38^\circ = \dfrac{h}{45}\), so \(h = 45 \tan 38^\circ = 45 \times 0.7813 = 35.16\) m.
Add the height of her eyes: \(35.16 + 1.6 = 36.76\) m.
Answer: the tower is 36.8 m high (3 s.f.)
Example 3 (Higher) In triangle \(ABC\), \(AB = 8.5\) cm, \(AC = 6.2\) cm and angle \(BAC = 54^\circ\). Find (a) \(BC\), (b) angle \(ABC\), (c) the area of the triangle.
(a) Two sides and the included angle are known, so use the cosine rule.
\(BC^2 = 8.5^2 + 6.2^2 - 2(8.5)(6.2)\cos 54^\circ = 72.25 + 38.44 - 105.4 \times 0.5878\)
\(BC^2 = 110.69 - 61.95 = 48.74\), so \(BC = 6.98\) cm (3 s.f.).
(b) Use the sine rule with the known pair \(BC\) and angle \(A\): \(\dfrac{\sin B}{6.2} = \dfrac{\sin 54^\circ}{6.981}\).
\(\sin B = \dfrac{6.2 \times 0.8090}{6.981} = 0.7185\), so \(B = 45.9^\circ\) (3 s.f.).
Angle \(B\) is opposite \(AC\), which is shorter than \(BC\), so \(B\) is smaller than \(A = 54^\circ\). The acute value is therefore the right one.
(c) Area \(= \tfrac12 \times 8.5 \times 6.2 \times \sin 54^\circ = 26.35 \times 0.8090 = 21.3\) cm² (3 s.f.).
Answer: (a) 6.98 cm (b) \(45.9^\circ\) (c) 21.3 cm²
Common errors
- Common error: adding the squares when finding a shorter side. For a shorter side you subtract: \(a^2 = c^2 - b^2\).
- Common error: calculator in radian mode, which gives completely wrong values for angles in degrees.
- Common error: labelling opposite and adjacent from the wrong angle. Mark the angle first, then label the sides from it.
- Common error: using the sine rule when the known angle is not opposite a known side, or using the cosine rule with an angle that is not between the two sides.
- Common error: writing a bearing with fewer than three figures or measuring it anticlockwise.
Exam tip: Before you press any buttons, write the equation with the numbers in it, such as \(\tan 38^\circ = \dfrac{h}{45}\). Then rearrange. Rounding only at the final step keeps your answer accurate.
Check yourself
- A right-angled triangle has hypotenuse 13 cm and one side 5 cm. Find the other side and the smaller angle.
Answer
Other side \(= \sqrt{169 - 25} = \sqrt{144} = 12\) cm. The smaller angle is opposite the 5 cm side: \(\sin\theta = \tfrac{5}{13}\), so \(\theta = 22.6^\circ\) (3 s.f.). - A cuboid measures 6 cm by 4 cm by 3 cm. Find the length of its space diagonal (Higher).
Answer
\(\sqrt{6^2 + 4^2 + 3^2} = \sqrt{36 + 16 + 9} = \sqrt{61} = 7.81\) cm (3 s.f.). - A kite on a taut string 80 m long makes an angle of \(35^\circ\) with level ground. How high is the kite? Ignore the height of the person holding the string.
Answer
Height is opposite the angle and the string is the hypotenuse: \(h = 80 \sin 35^\circ = 80 \times 0.5736 = 45.9\) m (3 s.f.). - Find the exact value of \(\sin 60^\circ \times \cos 30^\circ\) (Higher).
Answer
\(\dfrac{\sqrt3}{2} \times \dfrac{\sqrt3}{2} = \dfrac{3}{4}\).