Pythagoras' theorem and trigonometry

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Lesson notes

Pythagoras' theorem

In a right-angled triangle, the hypotenuse c is the longest side, opposite the right angle: a² + b² = c².
To find the hypotenuse, add the squares and square root. To find a shorter side, subtract: b² = c² - a².
A triangle with sides 3, 4, 5 is right-angled because 9 + 16 = 25 (a Pythagorean triple). Others: 5, 12, 13 and 8, 15, 17.

A 3–4–5 right triangle

Trigonometric ratios (SOHCAHTOA)

For an angle \θ: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent.
For an acute angle θ in a right triangle: the hypotenuse is opposite 90°, the opposite side is across from θ, and the adjacent side is the other side next to θ.
To find a side, rearrange the ratio; to find an angle use the inverse function, e.g. \θ = \sin^-1\left(\frac{\text{opp}}{\text{hyp}}\right).

Sides relative to angle θ

Exact values

You should know: \sin 30° = \frac{1}{2}, \cos 60° = \frac{1}{2}, \tan 45° = 1, \sin 45° = \cos 45° = \frac{\√{2}}{2}, \sin 60° = \cos 30° = \frac{\√{3}}{2}, \tan 60° = \√{3}, \tan 30° = \frac{1}{\√{3}}.

Exact values

Applications

Angles of elevation and depression are measured from the horizontal.
Bearings are measured clockwise from north; draw a sketch and add right-angled triangles.
Check calculator mode is in degrees.
Assume a wall or tower is vertical and the ground is horizontal only when stated or justified. Add the observer’s eye height when the angle is measured above eye level.

Two perpendicular journeys

Slides

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Practice questions

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1.Which side of a right-angled triangle is the hypotenuse?
Easy
- Athe shortest side
- Bthe longest side, opposite the right angle
- Cthe side next to the angle
- Dany side
2.A right-angled triangle has shorter sides 3 and 4. What is the hypotenuse?
Easy
- A5
- B7
- C12
- D25
3.For an acute angle θ in a right-angled triangle, what does sin θ equal?
Easy
- Aadj ÷ hyp
- Bopp ÷ hyp
- Copp ÷ adj
- Dhyp ÷ opp
4.For an acute angle θ in a right-angled triangle, what does tan θ equal?
Easy
- Aopp ÷ hyp
- Badj ÷ hyp
- Copp ÷ adj
- Dadj ÷ opp
5.A right-angled triangle has hypotenuse 13 and one side 5. What is the other side?
Easy
- A12
- B8
- C18
- D144
6.Which is a Pythagorean triple?
Easy
- A2, 3, 4
- B5, 12, 13
- C6, 7, 8
- D4, 5, 7
7.What is \sin 30°?
Easy
- A\frac{1}{2}
- B\frac{\√{3}}{2}
- C1
- D\frac{\√{2}}{2}
8.What is \cos 60°?
Easy
- A\frac{1}{2}
- B\frac{\√{3}}{2}
- C1
- D0

More questions

9.What is \tan 45°?
Easy
- A0
- B\frac{1}{2}
- C1
- D\√{3}
10.What is \sin 90°?
Easy
- A0
- B1
- C\frac{1}{2}
- Dundefined
11.A right-angled triangle has hypotenuse 10 and an angle of 30°. What is the side opposite the angle?
Medium
- A5
- B8.66
- C17.3
- D3
12.In a right-angled triangle the angle is 40° and the adjacent side is 8 cm. What is the opposite side (1 d.p.)?
Medium
- A6.7 cm
- B5.1 cm
- C9.5 cm
- D10.4 cm
13.A right-angled triangle has opposite 6 and hypotenuse 10. Find the angle (1 d.p.).
Medium
- A36.9°
- B53.1°
- C30.0°
- D59.0°
14.A right-angled triangle has opposite 5 and adjacent 12. Find the angle (1 d.p.).
Medium
- A22.6°
- B67.4°
- C25.0°
- D42.0°
15.A 10 m ladder rests against a vertical wall on horizontal ground, making an angle of 70° with the ground. How high does it reach, to 1 decimal place?
Medium
- A9.4 m
- B3.4 m
- C10.6 m
- D6.4 m
16.From a point on horizontal ground 50 m from the foot of a vertical tower, the angle of elevation to its top is 35°. Neglect observer eye height. How tall is the tower, to 1 decimal place?
Medium
- A35.0 m
- B28.7 m
- C61.0 m
- D40.9 m
17.Is a triangle with sides 7, 24 and 25 right-angled?
Medium
- Ayes
- Bno
- Conly if 24 is the hypotenuse
- Dcannot tell
18.Find the diagonal of a rectangle 8 cm by 15 cm.
Medium
- A17 cm
- B23 cm
- C12 cm
- D120 cm
19.Find the length of the side marked x: hypotenuse 12, angle 25°, x adjacent (1 d.p.).
Medium
- A10.9
- B5.1
- C25.7
- D13.2
20.Find x if \cos x = 0.5, with x acute.
Medium
- A60°
- B30°
- C45°
- D90°
21.A ship sails 10 km on a bearing of 090° then 24 km on a bearing of 180°. How far is it from the start?
Medium
- A26 km
- B34 km
- C14 km
- D22 km
22.What is the exact value of \tan 60°?
Medium
- A\√{3}
- B\frac{1}{\√{3}}
- C\frac{\√{3}}{2}
- D1
23.What is the exact value of \cos 45°?
Medium
- A\frac{\√{2}}{2}
- B\frac{1}{2}
- C1
- D\frac{\√{3}}{2}
24.Bearings are measured:
Easy
- Aanticlockwise from east
- Bclockwise from north
- Cclockwise from east
- Danticlockwise from north
25.An angle of depression is measured from:
Easy
- Athe vertical
- Bthe horizontal
- Cthe ground
- Dthe hypotenuse
26.Your calculator gives a nonsense answer for \sin 30°. The likeliest cause is that it is in:
Medium
- Aradians mode
- Bdegrees mode
- Cstatistics mode
- Dtable mode
27.In a right-angled triangle with angle \θ, which ratio uses the hypotenuse and the adjacent side?
Medium
- Asin
- Bcos
- Ctan
- Dnone
28.Pythagoras' theorem only applies to right-angled triangles.
Easy
True or false?
29.\sin 30° = \frac{1}{2}.
Easy
True or false?
30.The hypotenuse is always the shortest side.
Easy
True or false?
31.\tan\θ = \frac{\text{opposite}}{\text{adjacent}}.
Easy
True or false?
32.3, 4, 6 is a Pythagorean triple.
Medium
True or false?
33.Which are Pythagorean triples?
Medium
- A3, 4, 5
- B5, 12, 13
- C6, 7, 8
- D8, 15, 17
- E2, 3, 4
34.Which exact values are correct?
Medium
- A\sin 30° = \frac{1}{2}
- B\cos 60° = \frac{1}{2}
- C\tan 45° = 1
- D\sin 60° = \frac{1}{2}
- E\cos 30° = 1
35.A 6 m ladder rests against a vertical wall on horizontal ground, with its foot 2 m from the wall. How high up the wall does it reach? Give your answer to 2 decimal places.
Medium

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