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6a · E6.1–E6.3

Pythagoras and right-angled triangles

Topic 6 Trigonometry · Core and Extended · Papers 1–4

🎯What you need to be able to do

  • Use Pythagoras’ theorem to find a missing side of a right-angled triangle.
  • Use sine, cosine and tangent to find sides and angles in right-angled triangles, including in bearings problems.
  • Solve problems with angles of elevation and depression, and know the perpendicular is the shortest distance to a line EXTENDED.
  • Know the exact values of sin and cos of 0°, 30°, 45°, 60°, 90° and tan of 0°, 30°, 45°, 60° EXTENDED.

📚The mathematics

Pythagoras’ theorem

In a right-angled triangle with hypotenuse \(c\) (the longest side, opposite the right angle): \( a^{2} + b^{2} = c^{2} \). To find the hypotenuse, add the squares; to find a shorter side, subtract. A quick check: the hypotenuse must be the longest side.

Sine, cosine and tangent

\( \sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}} \)
\( \cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}} \)
\( \tan\theta = \dfrac{\text{opposite}}{\text{adjacent}} \)
A right-angled triangle with the angle theta marked at the bottom left. The side opposite theta is labelled opposite, the side next to theta (not the hypotenuse) is labelled adjacent, and the side opposite the right angle is labelled hypotenuse.
Label the sides from the angle you are using: opposite it, next to it, and the hypotenuse.

Label the sides, pick the ratio that uses the two sides involved (one known, one wanted), then rearrange. To find an angle, use the inverse: \( \theta = \tan^{-1}\left(\tfrac{\text{opp}}{\text{adj}}\right) \). Give angles to 1 decimal place, and make sure the calculator is in degrees.

Elevation, depression and bearings

EXTENDED The angle of elevation is measured up from the horizontal; the angle of depression down from the horizontal. Because the two horizontals are parallel, the angle of depression from \(A\) to \(B\) equals the angle of elevation from \(B\) to \(A\) (alternate angles). The shortest distance from a point to a line is along the perpendicular.

A cliff 60 metres high with an observer at the top and a boat at sea level 184.7 metres away. From the top, a horizontal dashed line and the line of sight down to the boat make an angle of depression of 18 degrees; at the boat, the angle of elevation to the top of the cliff is also 18 degrees.
Practice question 5: the angle of depression from the cliff top equals the angle of elevation from the boat.

In bearings problems, draw a north line at each point and look for right-angled triangles: a journey of \(d\) km on a bearing \( \theta \) moves \( d\cos\theta \) north and \( d\sin\theta \) east.

Exact values EXTENDED

Two triangles for exact values. Left: an isosceles right-angled triangle with shorter sides 1 and hypotenuse root 2, and angles of 45 degrees. Right: half of an equilateral triangle of side 2, with sides 1, root 3 and 2, and angles of 30 and 60 degrees.
Two triangles hold every exact value: \( \sin 45^{\circ} = \tfrac{1}{\sqrt{2}} \), \( \sin 30^{\circ} = \tfrac{1}{2} \), \( \cos 30^{\circ} = \tfrac{\sqrt{3}}{2} \), \( \tan 60^{\circ} = \sqrt{3} \), and so on.
\( \sin 0^{\circ} = 0 \), \( \sin 90^{\circ} = 1 \)
\( \cos 0^{\circ} = 1 \), \( \cos 90^{\circ} = 0 \)
\( \tan 0^{\circ} = 0 \), \( \tan 45^{\circ} = 1 \)
\( \tan 30^{\circ} = \tfrac{1}{\sqrt{3}} \), \( \tan 60^{\circ} = \sqrt{3} \)

✏️Worked example

(a) (Non-calculator.) A 6.5 m ladder leans against a wall with its foot 2.5 m from the wall. How high up the wall does it reach? [2] (b) (Calculator.) A right-angled triangle has hypotenuse 12 cm and an angle of \( 38^{\circ} \). Find the side opposite this angle. [2] (c) (Calculator.) The two shorter sides of a right-angled triangle are 5 cm and 9 cm. Find the smallest angle. [2] (d) (Non-calculator.) EXTENDED Find the exact value of \( \sin 60^{\circ} \times \tan 30^{\circ} \). [2]

(a) \( h = \sqrt{6.5^{2} - 2.5^{2}} = \sqrt{42.25 - 6.25} = \sqrt{36} = 6 \) m.

(b) \( \text{opp} = 12\sin 38^{\circ} = 7.39 \) cm.

(c) The smallest angle is opposite the smallest side: \( \tan^{-1}\left(\tfrac{5}{9}\right) = 29.1^{\circ} \).

(d) \( \tfrac{\sqrt{3}}{2} \times \tfrac{1}{\sqrt{3}} = \tfrac{1}{2} \).

Check it. (a) 2.5, 6, 6.5 is the 5, 12, 13 triangle halved ✓. (b) \( \sin 38^{\circ} \approx 0.62 \), so the side should be a bit more than half the hypotenuse ✓.
Adding when you should subtract. In (a), \( \sqrt{6.5^{2} + 2.5^{2}} = 6.96 \) would be longer than the ladder itself — impossible.

📝Practise

Questions in the style of the current papers. EXTENDED marks Extended-only content.

1. (Calculator.) The shorter sides of a right-angled triangle are 7.2 cm and 9.6 cm. Find the hypotenuse. [2]
\( \sqrt{7.2^{2} + 9.6^{2}} = \sqrt{144} = 12 \) cm.
2. (Calculator.) A right-angled triangle has an angle of \( 52^{\circ} \), with the adjacent side 8 cm. Find the opposite side. [2]
\( 8\tan 52^{\circ} = 10.2 \) cm.
3. (Calculator.) In a right-angled triangle the hypotenuse is 10 cm and the side adjacent to angle \(x\) is 7 cm. Find \(x\). [2]
\( \cos x = 0.7 \Rightarrow x = 45.6^{\circ} \).
4. (Calculator.) A ship sails 8 km on a bearing of \( 040^{\circ} \). How far north and how far east of its start is it? [3]
North \( 8\cos 40^{\circ} = 6.13 \) km; east \( 8\sin 40^{\circ} = 5.14 \) km.
5. (Calculator.) EXTENDED From the top of a 60 m cliff, the angle of depression of a boat is \( 18^{\circ} \). How far is the boat from the foot of the cliff? [2]
The angle of elevation from the boat is also \( 18^{\circ} \), so \( \tan 18^{\circ} = \tfrac{60}{d} \) and \( d = \tfrac{60}{\tan 18^{\circ}} = 185 \) m.
6. (Non-calculator.) EXTENDED Find the exact value of \( \cos 30^{\circ} \times \tan 60^{\circ} \). [2]
\( \tfrac{\sqrt{3}}{2} \times \sqrt{3} = \tfrac{3}{2} \).
7. (Calculator.) EXTENDED Triangle \(ABC\) has a right angle at \(C\), \( AC = 6 \) cm and \( BC = 8 \) cm. Find the shortest distance from \(C\) to the line \(AB\). [3]
\( AB = 10 \). The area is \( \tfrac{1}{2} \times 6 \times 8 = 24 \), and also \( \tfrac{1}{2} \times 10 \times d \), where \(d\) is the perpendicular distance. So \( d = 4.8 \) cm.

🔗Go deeper — other people’s work

These are external resources, not mine. If one stops working, tell me and everything above it on this page still stands.

  • Corbettmaths — SOHCAHTOA, elevation and depression
  • Your calculator manual — check it is in degree mode (a small D or DEG on the display)