Trigonometry is the study of triangles.


The sides of  any right angled triangle can be labelled
Hypotenuse, Adjacent and Opposite.

The hypotenuse is always opposite the right angle.

It is the longest side of the triangle.

The adjacent is the side that forms part of the required angle.
The opposite is the side directly across from the required angle.

1

or from the top angle,

hyp2

Ratios

The ratio of these sides is given special names:
sine, cosine and tangent.


These are shortened to sin, cos and tan

1

To find an angle or side, follow this recipe:-

       Always draw a sketch


Using Tan to find angle


Example

An aircraft is 73m above a building and 200m from its touchdown point.
Calculate angle θ˚, the glide path of the aircraft.
Give your answer correct to one decimal place.

2


2

2


The angle of the glide path of the aircraft is 20.1˚ (1dp)


Using Tan to find side

Example

An aircraft is  200m from its touchdown point, on a glide path of 30˚.
How high above the ground is the aircraft?
Give your answer correct to one decimal place.


2


2

3


The height of the aircraft is 115.5m (1dp)


Using Sine to find side


Example

A skier is racing down a 150m long ramp which has an angle of inclination of 30°.
How high above the ground is the starting flag?


2

 


2

 

4


The starting flag is 50m above ground.


Using Sine to find angle

Example

A skier is racing down a ramp which is 200m long.
The start of the ramp is 100m above ground level.
What is the value of  θ˚, the angle of inclination of the ramp?

2


20

5

The  angle of inclination of  the ramp is 30˚.


Using Cos to find side

Example

A ship is at anchor. The chain is 150 m long and makes an angle of 30˚ from the anchor point to the seabed. The anchor point is 5m above the water level. How deep in the water is the anchor ? Give your answer to the nearest metre.

 

2

 

22

6

 


Using Cos to find angle

Example

A ship is taut at anchor. The chain is 150 m long and lies  75m from the anchor point. What is the value of θ˚, the  angle from the seabed  to the anchor point at the ship’s bow ?
222


2

7


Which ratio should be used ?

The following recipe can be used for trig problem solving questions :-

  1. Find and sketch the triangle
  2. Mark the right angle
  3. Identify and mark angle to be used / found
  4. Label Opposite, Hypotenuse and Adjacent
  5. Write out


2

  1. Two Tick Test:-
         Tick what you know   
         Tick what you want  
    Look for two ticks and use that ratio
  2. Write out ratio and solve.
  3. Write down solution

Example

2
How high up the wall is the ladder ?


(Give your answer to 2d.p.)

 


2

 

Tick what you know     Adjacent
Tick what you want       Opposite


2


Look for two ticks  and use that ratio.

  ( Here, use tan)


8  

The ladder is 1.73m (2dp) up the wall.


Finding the area of a triangle

What is the area of this triangle?


2


Method 1
Draw perpendicular BD.
Use basic trigonometry to find length of BD.
(This is the height of the triangle.)

Then use Area  = 1/2 x base x perpendicular  height


So,


2

9

A = 1/2 x base x height
A = 1/2 x 12 x 9.511
A = 57.063391
A = 57.1 cm2   (1 dp)


Method 2


Name sides opposite vertices with their lower case letters.


2

so


22


2 10

11




2 12

Example


A Hollywood film set is being constructed in the desert. The painter is approaching the  square based pyramid from behind a sand dune.
The base is 11m, the slant height is 9m and the angle between the base and slant height is 52°.

2

Calculate the area of the face of the pyramid shown.


2

13


The Sine Rule

What is the length of AC ?


2

2

This gives


   14 and     15 

So,

18

Using the letters,

16   and    17

     19


Which gives The Sine Rule


2 20

 


Example


A helicopter is approaching an oilrig. The distance from the helicopter to the oilrig is 10 km.
 From the bridge of the tanker, the oil rig bears due East and the helicopter bears  040˚ at a distance of 15 km.

 

2

What is the bearing of the helicopter from the oil rig ?
   Give your answer correct to the nearest degree.


2
 

21

22

23

 


The Cosine Rule


What is the length of BC ?

2

2

3 

No  pairs – so can’t use Sine Rule

2

222 and 24

 

26

 

27

28


This is known as the Cosine Rule

2 29

 

Re-arranging gives

31

32


Example


A satellite is being tracked by an observatory telescope and a transmitter.
When the angle of elevation of the telescope is 40˚, the satellite is known to be a distance of 6500 km from the observatory.


The distance between the observatory and the  transmitter is 50 km, at a level height.

  What is the distance of the satellite  from the transmitter?
Give your answer to the nearest kilometer.


2


2


Use Cosine rule

33

        


      Example


The space shuttle is tied down by two 120 m long chains.
The distance between the tie down points  on the ground is 100 m.


What is the value of  angle θ˚ ? Give your answer correct to 1 decimal place.


2


2

34

 



Which rule to use


AAA – 3 angles

2

not   enough information

 

AAS – 2 angles, 1 side 

2 
Use Sine Rule

 


ASS – 1 angle ( not included) , 2 sides

2

Ambiguous case: Either Use Sine Rule twice, or  use Cosine Rule with quadratic formula.

Using the Sine Rule:

36

35

Using the Cosine Rule

37

Now solve using the quadratic formula

39

so

40

or

41

Discard the negative value, since the length cannot be negative.

38

 

SAS –  Included angle, 2 sides 

2

Use  Cosine Rule 


 SSS – 3 sides 
22222
Use  Cosine Rule

© Alexander Forrest