SINE AND COSINE RULES

TIP: When doing these sort of problems, remember:

a) Whenever you need to work out lengths of sides or angles in a triangle which does not have a right-angle, you need to use either the Sine or Cosine rules. (Sin, Cos, Tan and Pythagoras' Theorem do not work for triangles without a right-angle).

b)

c) Note that the Sine and Cosine rules are commonly seen on formulae sheets in exams - so don't learn them if you don't need to.

Solve the following:

1.Calculate the angle x in the triangle below:

 

2. Calculate the length y of the side in the triangle below:

 

3.  In the quadrilateral below, calculate the size of the angle ABC:

4. The picture below shows a farmer's field after it has been mapped out.
i) Find the length AC
ii) Find the angle ADC
iii) Find the area of the field

5.

Using the details given in the triangle above:

a) Show that 3x2 + 6x - 36 = 0
b) Solve the equation 3x2 + 6x - 36 = 0 and give your answer to 3 significant figures.
c) D is a point on the line AC such that the angle ADB = 100o. Calculate the length of BD.

6. A hiker starts her journey at point A. She notices a farm house at point C and works out its bearing is at 138o. She then walks for 5 kilometres and stops at point B. At point B the hiker looks again at the farm house and calculates its bearing now to be 200o. Calculate the distances AC and BC.

Here is a diagram of the hiker's path and bearing measurements:

7.

The diagram above represents air traffic control at point C. An aeroplane follows a direct path from A to B.

i) Calculate the distance the aeroplane flies between the points A and B

ii) At its shortest distance from air traffic control, the aeroplane is at the point X on the line AB. Calculate the distance CX.

8. A ship sails from harbour and travels 25km on a bearing of 30o before reaching a marker bouy. At this point the ship turns and follows a course on a bearing of 90o and travels for 32km until it reaches an island. On the return journey, the ship is able to take the most direct route back to the harbour.

i) Draw a diagram to represent both the outbound and return journeys.

ii) What is the total distance travelled by the ship?

Ok Here's the Answers