Sometimes you’ll get given a diagram of a triangle with two of the side lengths labelled, but not the angle. The question will ask you to find the angle, usually to a certain degree of accuracy. In these cases, start with a guess at the angle, and see if it works using the appropriate cos, sin or tan ratio. If it doesn’t, try increasing the angle. If this gets you closer keep going, otherwise try decreasing the angle.

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Find θ to the nearest degree: |

Solution |

If the diagram is roughly to scale, we can make an initial guess at what θ is – let’s say it’s about 40°. Now, to check whether that’s a good guess, we can use a trigonometric ratio. But which one to use? Well, relative to the θ angle, the ‘5’ side is the adjacent side, and the ‘6’ side is the hypotenuse. So we want the cos ratio: So what we are trying to do is find a value of θ that will make this equation true. Now, we’ve guessed that θ is 40°. We can use our calculator to work out what cos of 40° is: So our guess of 40° gives us an answer of 0.7660, which is smaller than 0.8333. Let’s try increasing our guess for θ to 41°: Uh-oh! 0.7547 is even further away from 0.8333, so
perhaps we should be Aha! We’re getting closer to 0.8333 now, so we should
continue That’s a lot closer to 0.8333, but we’re still a bit short. So let’s drop another 2 degrees: Now, 0.8387 is go back to 34°: So the 34° is 0.8290, which is cos 33°: cos 34°: So the result of cos 34° is a smaller distance from what
we want – 0.8333, so we’ll use it as our answer – θ = 34°. There is a
much faster way to do this question using the |

Handy Hint #1 - Application problems in trigonometry

One of the common questions you get in trigonometry is one about a ship, person or plane etc… travelling a certain distance at a certain angle. You’re then usually asked to find out how far in a certain direction, north for instance, the person is of where they started. There are a lot of variations on this type of question, but the general procedure for solving them is the same:

- Draw a diagram with as much information labelled on it as possible

- Work out what you’re trying to find, and label it on the diagram

- Work out whether you have enough information to finish the problem straightaway, or whether you’re going to need to find out some other values first before you can find the final answer

- Work through the problem. Each time you find out some more information, draw another diagram if possible and label it on the new diagram, or add it to the original diagram.

Bobby rows his boat across a wide river. He aims straight at the other bank, but because of the current, his actual path is a straight line 28° away from the shortest path across the river. The river is 250 metres wide. How far did Bobby actually travel? How much further downstream is he then if there had been no current? |

Solution |

First up, let’s draw a diagram. We need a river, 250
metres wide. We need to draw Bobby’s path across the river, which is a So in this diagram I have drawn the river by drawing two parallel straight lines horizontally – these lines represent the banks (sides) of the river. In between them is the water, where I’ve drawn the two paths across the river – the shortest possible path and the path that Bobby actually took. I’ve labelled these paths as well. I’ve put numerical information from the question into the
diagram – the river is 250 metres wide, and Bobby’s path is 28° off the
shortest path direction. I’ve also labelled the things that we want to try
and find. The distance Bobby actually travelled is labelled using a ‘d’.
The distance further downstream between where he would have been had he gone
straight across the river, and where he did go because of the current, is
labelled using a ‘h’. So we’ve got a diagram, now we need to find some answers. The central part of the diagram you should quickly realise is just a right-angled triangle. And we know one of the angles in it – 28°, and also one of the side lengths – 250 metres. This means we can use trigonometry to find the lengths of the other two sides – which happen to be the two things we’re trying to find out. First up, let’s find the length of ‘d’. Relative to the
angle 28°, I have labelled which sides are the adjacent, opposite and
hypotenuse. To find ‘d’, we want a trigonometric ratio that involves both
the unknown ‘d’, and the known side which is 250 metres long. These are the hypotenuse
and adjacent sides, so we want to use After a bit of rearranging, all we need to do now is use our calculator to work out what cos 28° is, and find d: Looking at the diagram, this distance seems to make sense. Let’s move on to finding ‘h’. Once again we want a trigonometric ratio that involves
the unknown side ‘h’ and the known 250 metre side (we use the 250 metre side
instead of the 283.1 metre side because it’s an Now we just use our calculator: Notice I’ve used |