Maths - Trigonometry and Bearings

Area of a triangle: To be able to calculate the area of a triangle, you need to know two sides and its included angle. Example: Find the area of triangle ABC.

The sine rule: We can use the sine rule when we're given the sizes of: two sides and one angle opposite to one of these sides to find the other angle one side and any two angles to find the size of another side Example 1: Find the size of angle R. Substitute the information from the diagramCross multiply (top right x bottom left = top left x bottom right)Use 'change side, change operation'. The '9' is multiplying on the left so when it goes to the other side of the equals sign, it does the opposite so divide.Use 'change side, change operation' to get R on its own.Moving sin to the other side becomes sin-1.Remember to press 'shift' then 'sin' to get 'sin-1' on your calculator and remember to close your bracket after the angle. Example 2 Find the length of YZ. Answer We know two angles and so can calculate the third angle in the triangle, ie  .Remember to close your bracket after  Therefore 

The cosine rule: Finding a side: The cosine rule can be rearranged to calculate either the size of a side or an angle.Use this formula when given the sizes of two sides and its included angle. Example: Find the length of BC. Answer As we are calculating the length of a side, we use the first formula. Finding an angle: Use this formula when given the sizes of all three sides. Example: Find the size of angle R. Answer As we are calculating the size of an angle, we use the second formula.

Bearings: Remember a bearing in mathematics is the angle in degrees measured clockwise from north. These are usually given as a three-figure bearing. For example, 30° clockwise from north is usually written as 030°. Example: A boat leaves the harbour and travels 9 km north west, then 12 km north. Calculate its distance and bearing from the harbour. Answer We need to sketch the problem first. Since the boat travels from the harbour on a NW direction then this tells us that it is travelling at a bearing of 315°. It then travels north, which means the harbour is now at a bearing of 135° from its position now since it is now SE.Looking at the diagram, we are asked to find an angle and a side, although the angle we have to find is outside the triangle.Let's find the side first, which is the distance the boat is from the harbour.We know two sides and included angle, so from the flow chart we saw previously, we know that we are using the cosine rule. (to nearest km)Now we can find the angle. Let's look again at the sketch. Let's look again at the sketch.If we find the angle labelled with a star then we can find our required angle by subtracting this answer from 180° since it is a supplementary angle (from a straight angle which adds up to 180°).From the flow chart from before, you can see that we can use either the sine rule or the cosine rule, so let's choose the sine rule. (to nearest degree)Therefore the angle we require is:  . So, the boat is now 19 km on a bearing of 160° from the harbour.

Trigonometry

Bearings