1. Whiteboardmaths.com
© 2004 All rights reserved 5 7 2 1

2. Trigonometry
Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.
Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it. ?

3. Trigonometry
Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.
Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it.
30o

4. Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.
Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it.
Trigonometry
35o

5. Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.
Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it.
Trigonometry
40o

6. What’s he going to do next?
Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.
Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it.
Trigonometry ?
What’s he going to do next?
45o

7. What’s he going to do next?
Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.
Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it.
Trigonometry
45o ?
What’s he going to do next?
324 m

8. Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.
Trigonometry
45o
324 m
324 m

9. 324 m Trigonometry Eiffel Tower Facts: Designed by Gustave Eiffel.
Completed in 1889 to celebrate the centenary of the French Revolution.
Intended to have been dismantled after the 1900 Paris Expo.
Took 26 months to build.
The structure is very light and only weighs tonnes.
pieces, 2½ million rivets.
1665 steps.
Some tricky equations had to be solved for its design.

10. Early Beginnings

11. Trigonometry (Three-angle-measure)
The Great Pyramid (Cheops) at Giza, near Cairo, one of the 7 wonders of the ancient word. (The only one still surviving).

12. Trigonometry
Sun’s rays casting shadows mid-afternoon
Sun’s rays casting shadows late afternoon
An early application of trigonometry was made by Thales on a visit to Egypt. He was surprised that no one could tell him the height of the 2000 year old Cheops pyramid. He used his knowledge of the relationship between the heights of objects and the length of their shadows to calculate the height for them. (This will later become the Tangent ratio.) Can you see what this relationship is, based on the drawings below?
Thales of Miletus 640 – 546 B.C. The first Greek Mathematician. He predicted the Solar Eclipse of 585 BC.
6 ft
9 ft
720 ft h
480 ft
Similar Triangles
Similar Triangles
Thales may not have used similar triangles directly to solve the problem but he knew that the ratio of the vertical to horizontal sides of each triangle was constant and unchanging for different heights of the sun. Can you use the measurements shown above to find the height of Cheops?

13. Trigonometry
Sun’s rays casting shadows mid-afternoon
Sun’s rays casting shadows late afternoon
An early application of trigonometry was made by Thales, on a visit to Egypt. He was surprised that no one could tell him the height of the 2000 year old Cheops pyramid. He used his knowledge of the relationship between the heights of objects and the length of their shadows to calculate the height for them. (This will later become the Tangent ratio.) Can you see what this relationship is, based on the drawings below?
Thales of Miletus 640 – 546 B.C. The first Mathematician. He predicted the Solar eclipse of 585 BC.
Similar Triangles
6 ft
9 ft
720 ft h
480 ft

14. Hipparchus of Rhodes 190-120 BC
Later, during the Golden Age of Athens (5C BC.), the philosophers and mathematicians were not particularly interested in the practical side of mathematics so trigonometry was not further developed. It was another 250 years or so, when the centre of learning had switched to Alexandria (current day Egypt) that the ideas behind trigonometry were more fully explored. The astronomer and mathematician, Hipparchus was the first person to construct tables of trigonometric ratios. Amongst his many notable achievements was his determination of the distance to the moon with an error of only 5%. He used the diameter of the Earth (previously calculated by Eratosthenes) together with angular measurements that had been taken during the total solar eclipse of March 190 BC.
Eratosthenes275 – 194 BC
Hipparchus of Rhodes BC
The library of Alexandria was the foremost seat of learning in the world and functioned like a university. The library contained manuscripts.

15. Early Applications of Trigonometryh
20o
25o x d
Early Applications of Trigonometry
Finding the height of a mountain/hill.
Finding the distance to the moon.
Constructing sundials to estimate the time from the sun’s shadow.

16. Historically trigonometry was developed for work in Astronomy and Geography. Today it is used extensively in mathematics and many other areas of the sciences.
Surveying
Navigation
Physics
Engineering

17. Trigonometry
The ideas behind trigonometry are based firmly on the previous work on similar triangles. In particular we are interested in similar right-angled triangles. A B
50o
40o
Explain why triangles A and B are similar.
Explain why triangles C and D are similar. C D
4 cm
3 cm
5 cm
8 cm
6 cm
10 cm A
Because they are equiangular.
50o
40o
Because corresponding sides are in proportion:
5/10 =
4/8 =
3/6 = ½
This means of course that A is an enlargement of B
C is enlargement of D by scale factor x 2

18. Trigonometry A C D
4 cm
3 cm
5 cm
8 cm
6 cm
10 cm
Corresponding sides are in proportion
5/10 = 4/8 = 3/6 = ½ A C D
4 cm
3 cm
5 cm
8 cm
6 cm
10 cm
Compare the ratio of any two sides in triangle C to the corresponding pair in triangle D. What do you notice?
The ratio of any two sides in one triangle is equal to the ratio of the corresponding pair in the other.
6/10 = 3/5 (= 0.6)
6/8 = 3/4 (= 0 .75)
8/10 = 4/5 (= 0.8)
This relationship is always true for similar right-angled triangles.

19. Convention for labelling triangles.
6/10 = 3/5 (= 0.6)
6/8 = 3/4 (= 0 .75)
8/10 = 4/5 (= 0.8) A C D
4 cm
3 cm
5 cm
8 cm
6 cm
10 cm
In similar right-angled triangles the ratios of any pair of sides in one triangle is equal to the ratio of the corresponding pair in all others. It is this idea that forms the basis for trigonometry.
Convention for labelling triangles. A B C
Angles denoted by CAPITAL letters.
Sides opposite a given angle use the same letter but in lower case. c
Side c opposite angle C b
Side b opposite angle B a
Side a opposite angle A

20. Convention for labelling triangles.
6/10 = 3/5 (= 0.6)
6/8 = 3/4 (= 0 .75)
8/10 = 4/5 (= 0.8) A C D
4 cm
3 cm
5 cm
8 cm
6 cm
10 cm
In similar right-angled triangles the ratios of any pair of sides in one triangle is equal to the ratio of the corresponding pair in all others. It is this idea forms the basis for trigonometry.
Convention for labelling triangles.
Angles denoted by CAPITAL letters.
Sides opposite a given angle use the same letter but in lower case. P Q R r
Side r opposite angle R q
Side q opposite angle Q p
Side p opposite angle P

21. Convention for naming sides.
6/10 = 3/5 (= 0.6)
6/8 = 3/4 (= 0 .75)
8/10 = 4/5 (= 0.8) A C D
4 cm
3 cm
5 cm
8 cm
6 cm
10 cm
In similar right-angled triangles the ratios of any pair of sides in one triangle is equal to the ratio of the corresponding pair in all others. It is this idea forms the basis for trigonometry.
Convention for naming sides. A B C
65o
The side opposite the right-angle is called the hypotenuse.
hypotenuse
The side opposite a given angle is called the opposite side.
adjacent
opposite
The side next to (or adjacent to) a given angle is called the adjacent side.

22. Convention for naming sides.
6/10 = 3/5 (= 0.6)
6/8 = 3/4 (= 0 .75)
8/10 = 4/5 (= 0.8) A C D
4 cm
3 cm
5 cm
8 cm
6 cm
10 cm
In similar right-angled triangles the ratios of any pair of sides in one triangle is equal to the ratio of the corresponding pair in all others. It is this idea forms the basis for trigonometry.
Convention for naming sides.
The side opposite a given angle is called the opposite side.
The side opposite the right-angle is called the hypotenuse
The sides next to (or adjacent to) a given angle is called adjacent side. A B C
25o
opposite
hypotenuse
adjacent

23. Convention for naming sides.
6/10 = 3/5 (= 0.6)
6/8 = 3/4 (= 0 .75)
8/10 = 4/5 (= 0.8) A C D
4 cm
3 cm
5 cm
8 cm
6 cm
10 cm
In similar right-angled triangles the ratios of any pair of sides in one triangle is equal to the ratio of the corresponding pair in all others. It is this idea forms the basis for trigonometry.
Convention for naming sides. A B C
38o
The side opposite a given angle is called the opposite side.
The side opposite the right-angle is called the hypotenuse
The sides next to (or adjacent to) a given angle is called adjacent side.
opposite
adjacent
hypotenuse

24. The Trigonometric RatiosB C
hypotenuse
opposite
adjacent
Make up a Mnemonic! S O C H A T

25. S O C H A T
Make up a Mnemonic!
Harry U R A R Y T E I S W N P L E

26. The Trigonometric Ratios (Finding an unknown side).
Use your worksheet and a ruler to measure as accurately as possible the sides of each triangle above. Find approximate values for the sine, cosine and tangent ratios for an angle of 30o.
True Values (2 dp)
The fixed values of these ratios were calculated for every angle and stored in a table of sines, cosines and tangents. Nowadays a calculator computes each value instead. Because these values are constant we can use them to find unknown sides.
To Trig Tables >
Sin 30o = 0.50
Cos 30o = 0.87
Tan 30o = 0.58

27. The Trigonometric Ratios (Finding an unknown side).
Sin 30o = 0.50
Cos 30o = 0.87
Tan 30o = 0.58
True Values (2 dp)
The Trigonometric Ratios (Finding an unknown side).
30o
For example, anytime we come across a right-angled triangle containing an angle of 30o we can find an unknown side if we are given the value of one other. h
30o
75 m

28. The Trigonometric Ratios (Finding an unknown side).
Example 1. In triangle ABC find side CB.
70o A B C
12 cm
Diagrams not to scale. S O H C A T
Opp
Example 2. In triangle PQR find side PQ.
22o P Q R
7.2 cm S O H C A T
Example 3. In triangle LMN find side MN.
75o L M N
4.3 m S O H C A T

29. The Trigonometric Ratios (Finding an unknown angle).
Anytime we come across a right-angled triangle containing 2 given sides we can calculate the ratio of the sides then look up (or calculate) the angle that corresponds to this ratio.
Sin 30o = 0.50
Cos 30o = 0.87
Tan 30o = 0.58
True Values (2 dp) xo
43.5 m
75 m S O H C A T
30o

30. The Trigonometric Ratios (Finding an unknown angle).
Example 1. In triangle ABC find angle A. A B C
12 cm
11.3 cm
Diagrams not to scale. S O H C A T
Sin-1(11.3  12) =
Key Sequence S O H C A T
Example 2. In triangle LMN find angle N. L M N
4.3 m
1.2 m
Tan-1(4.3  1.2) =
Key Sequence
Example 3. In triangle PQR find angle Q. P Q R
7.2 cm
7.8 cm S O H C A T
Cos-1(7.2  7.8) =
Key Sequence

31. Applications of Trigonometry
A boat sails due East from a Harbour (H), to a marker buoy (B), 15 miles away. At B the boat turns due South and sails for 6.4 miles to a Lighthouse (L). It then returns to harbour. Make a sketch of the trip and calculate the bearing of the harbour from the lighthouse to the nearest degree. H B L
15 miles
6.4 miles
SOH CAH TOA

32. Applications of Trigonometry
A 12 ft ladder rests against the side of a house. The top of the ladder is 9.5 ft from the floor. Calculate the angle that the foot of ladder makes with the ground.
Applications of Trigonometry Lo
SOH CAH TOA

33. Applications of Trigonometry
An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 430 miles North to a point P before turning left and flying for 570 miles to a second point Q, West of W. It then returns to base.
(a) Make a sketch of the flight.
(b) Find the bearing of Q from P.
Applications of Trigonometry
Not to Scale P
570 miles W
430 miles Q
SOH CAH TOA

34. Angles of Elevation and Depression.
An angle of elevation is the angle measured upwards from a horizontal to a fixed point. The angle of depression is the angle measured downwards from a horizontal to a fixed point.
Horizontal
Horizontal
25o
Angle of depression
Explain why the angles of elevation and depression are always equal.
25o
Angle of elevation

35. Applications of Trigonometry
A man stands at a point P, 45 m from the base of a building that is 20 m high. Find the angle of elevation of the top of the building from the man.
Applications of Trigonometry
20 m
45 m P
SOH CAH TOA

36. A 25 m tall lighthouse sits on a cliff top, 30 m above sea level
A 25 m tall lighthouse sits on a cliff top, 30 m above sea level. A fishing boat is seen 100m from the base of the cliff, (vertically below the lighthouse). Find the angle of depression from the top of the lighthouse to the boat.
100 m
55 m D C D
Or more directly since the angles of elevation and depression are equal.
SOH CAH TOA

37. A 22 m tall lighthouse sits on a cliff top, 35 m above sea level
A 22 m tall lighthouse sits on a cliff top, 35 m above sea level. The angle of depression of a fishing boat is measured from the top of the lighthouse as 30o. How far is the fishing boat from the base of the cliff?
x m
57 m
30o
60o
Or more directly since the angles of elevation and depression are equal.
30o
SOH CAH TOA

38. The origins of trigonometry are closely tied up with problems involving circles. One particular problem is that of finding the lengths of chords subtended by different angles at the centre of a circle. The Arabs called the half chord (“ardha-jya”). This became mis-interpreted and mis-translated over the centuries and eventually ended up as “sinus” in Latin, meaning cove or bay. Other derivations include: bulge, bosom, sinus, cavity, nose and skull. The cosinus simply means the compliment of the sinus, since SinA = Cos (90 – A) (Sin 60 = Cos 30, Sin 70 = Cos 20 etc)
Chord M
Angular Bisector
Radius P O Q
Half Chord PM (Sinus)
OM (Cosinus)

39. The following diagrams show the relationships between the 3 trigonometric ratios for a circle of radius 1 unit. Tangent means “To touch”
Radius P O Q
Half Chord PM
Sin  = O/H = PM/1 = PM
Chord
Cos  = A/H = OM/1 = OM M
Angular Bisector o
Tan  = PT/1 = PT
Tangent T o P O 1 M P M O 1  P O T  1
Sinus
Cosinus M O P 1 

40. The Trigonometric Ratios (Finding an unknown side).
Use your worksheet and a ruler to measure as accurately as possible the sides of each triangle above. Find approximate values for the sin, cos and tan ratios for an angle of 30o.
Sin 30o =
Cos 30o =
Tan 30o =
Estimated Values
Worksheet 1

41. Trig Tables
Click to go back

42. The Trigonometric Ratios (Finding an unknown side).
Example 1. In triangle ABC find side CB.
70o A B C
12 cm
Opp
Example 2. In triangle PQR find side PQ.
22o P Q R
7.2 cm
Example 3. In triangle LMN find side MN.
75o L M N
4.3 m
Worksheet 2

43. The Trigonometric Ratios (Finding an unknown angle).
Example 1. In triangle ABC find angle A. A B C
12 cm
11.3 cm
The Trigonometric Ratios (Finding an unknown angle).
Example 2. In triangle LMN find angle N. L M N
4.3 m
Example 3. In triangle PQR find angle Q. P Q R
7.2 cm
7.8 cm
1.2 m
Worksheet 3