Right-Angled Triangles (Pythagoras & Trigonometry) Notes

Cambridge IGCSE Maths: Revision notes

Key facts

Pythagoras: $a^2+b^2=c^2$ , where $c$ is the hypotenuse, the longest side opposite the right angle.
SOH CAH TOA: $\sin\theta=\dfrac{\text{opp}}{\text{hyp}}$ , $\cos\theta=\dfrac{\text{adj}}{\text{hyp}}$ , $\tan\theta=\dfrac{\text{opp}}{\text{adj}}$ .
To find an angle, use the inverse function ( $\sin^{-1}$ , $\cos^{-1}$ , $\tan^{-1}$ ).
Keep full calculator values until the last line, then give angles to 1 decimal place.
The shortest distance from a point to a line is the perpendicular (Extended).

Pythagoras' theorem

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

The hypotenuse is the longest side, opposite the right angle. To find the hypotenuse, add the squares and take the square root: $c=\sqrt{a^2+b^2}$ . To find a shorter side, subtract: $a=\sqrt{c^2-b^2}$ .

A right-angled triangle with sides 6 cm and 8 cm: the hypotenuse is 10 cm

Pythagoras $a^2+b^2=c^2$

Worked example

A triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.

1
Write the theorem: $c^2=a^2+b^2$ .
2
Substitute: $c^2=6^2+8^2=36+64=100$ .
3
Square root: $c=\sqrt{100}$ .

The hypotenuse is 13 cm and one shorter side is 5 cm. How long is the other side?

Sine, cosine and tangent

Each ratio links an angle to two sides, named relative to that angle.

For an acute angle $\theta$ , the opposite side is across from $\theta$ , the adjacent side is next to $\theta$ (not the hypotenuse) and the hypotenuse is opposite the right angle. Choose the ratio that connects the side you know with the side or angle you want.

Sides named relative to the angle θ

sin θ $\dfrac{\text{opp}}{\text{hyp}}$
cos θ $\dfrac{\text{adj}}{\text{hyp}}$
tan θ $\dfrac{\text{opp}}{\text{adj}}$

The side opposite θ is 3 cm and the hypotenuse is 5 cm. What is sin θ?

Solving 2D problems

Pick the two sides involved, choose the matching ratio, then rearrange.

To find a missing side, decide which two sides are involved and use the matching ratio (or Pythagoras if no angle is involved). To find a missing angle, use the inverse function, such as $\theta=\sin^{-1}\!\left(\dfrac{\text{opp}}{\text{hyp}}\right)$ . Bearing problems often make right-angled triangles between compass directions, so sketch one first. Give angles to one decimal place.

1
Label
Mark opp, adj, hyp from the angle
2
Choose
Pythagoras, or SOH, CAH or TOA
3
Rearrange
Make the unknown the subject
4
Calculate
Keep full values, then round

Choosing the method

Worked example

Hypotenuse 12 cm and an angle of 35°. Find the side opposite the angle.

1
Opposite and hypotenuse, so use sine: $\sin35°=\dfrac{x}{12}$ .
2
$x=12\sin35°$ .

Worked example

Sides next to the right angle are 5 cm (opposite θ) and 8 cm (adjacent). Find θ.

1
Opposite and adjacent, so use tangent: $\tan\theta=\dfrac58$ .
2
$\theta=\tan^{-1}(0.625)$ .

Worked example

A ship sails 8 km east, then 6 km north. How far is it from the start, and on what bearing?

1
Distance: $\sqrt{8^2+6^2}=10$ km.
2
Angle from north towards east: $\tan\theta=\dfrac86$ , so $\theta\approx53.1°$ .

$\tan\theta=0.625$ . Which gives θ to 1 decimal place?

Shortest distance

The shortest distance from a point to a line is along the perpendicular.

The perpendicular distance from a point to a line is the shortest possible. Any slanted route is the hypotenuse of a right-angled triangle, so it is longer. This justifies using one particular right-angled triangle in minimum-distance problems.

P is 3 cm from the line along the perpendicular; every other route is longer

How is the shortest distance from a point to a straight line measured?

Elevation and depression

Both are measured from the horizontal, upwards for elevation and downwards for depression.

The angle of elevation is measured upwards from the horizontal to a point above. The angle of depression is measured downwards from the horizontal to a point below. The two horizontals are parallel, so by alternate angles the elevation from one point equals the depression from the other.

Angle of depression from the top of a 20 m cliff to a boat

Worked example

From the top of a 20 m cliff, the angle of depression to a boat is 15°. Find the horizontal distance to the boat.

1
By alternate angles, the angle at the boat is also 15°.
2
The cliff is opposite and the distance $d$ is adjacent: $\tan15°=\dfrac{20}{d}$ .
3
$d=\dfrac{20}{\tan15°}$ .

A tower is 30 m tall. You stand 40 m from its base. What is the angle of elevation of the top?

Try an exam question

A ship sails 8 km east and then 6 km north. (a) Calculate how far the ship is from its starting point. (b) Calculate the bearing of the ship from its starting point, to 1 decimal place.

[4 marks]

That's the notes covered.

Carry on to the next subtopic.