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4.10 G) Trigonometry Word Problems
4.10 G) Trigonometry Word Problems
Whenever we are given a word problem that involves anything visual, we should always draw it out. This is because by drawing the word problem out, we have an idea of what is happening. The sketch does not have to be extremely accurate; it just needs to give us a rough idea of what is happening.
A common exam question is where you are looking at an object at an angle. If we are looking up at an object, the angle between the horizontal (directly straight) and the line of vision/ site is known as the angle of elevation.
A common exam question is where you are looking at an object at an angle. If we are looking up at an object, the angle between the horizontal (directly straight) and the line of vision/ site is known as the angle of elevation.
If we are looking down at an object, the angle between the horizontal and the line of vision/ sight is known as the angle of depression.
Let’s have a few different examples
Example 1
I am lying on the floor looking up at the top of a tree. My eyes are 5m away from the stump of the tree and they are 12m away from the top of the tree. What is the angle of elevation? Give your answer to the nearest degree.
The first step in answering this question is to draw a quick sketch to represent the information that we are given/ are looking for.
I am lying on the floor looking up at the top of a tree. My eyes are 5m away from the stump of the tree and they are 12m away from the top of the tree. What is the angle of elevation? Give your answer to the nearest degree.
The first step in answering this question is to draw a quick sketch to represent the information that we are given/ are looking for.
From the above sketch, we can see that the adjacent is 5m (the distance between my eye and the base of the tree). Also, the hypotenuse is 12m (the distance between my eye and the top of the tree).
We now need to find out which formula triangle we need to use. We are given the hypotenuse and the adjacent. This means that we will be using CAH.
We now need to find out which formula triangle we need to use. We are given the hypotenuse and the adjacent. This means that we will be using CAH.
We are looking for the angle, which means that we will cover up C, which tells us that cos of the unknown angle is equal to the adjacent divided by the hypotenuse.
We want to find the size of the angle, which means that we need to take the inverse of cos from both sides; we cos-1 both sides.
The question asks us to give our answer to the nearest degree.
3 is less than 5, which means that we round down. Therefore, the angle that I am looking up at is 65° to the nearest degree.
Example 2
I am standing on the top of a cliff that is 20m above sea level (assume that I have no height; I am looking from 20m above sea level). I am looking down at a boat and the angle of depression is 35°. How far is the boat away from the cliff? Give your answer to 2 decimal places.
The first step is to create a quick sketch. The angle of depression is the difference between the horizontal and the line of vision. I am standing 20m above sea level. The horizontal will always be 20m above sea level. Therefore, I am going to create a right-angle triangle with the horizontal, line of vision and the height above sea level at the right of the diagram rather than at the left. We get the right-angle triangle that is given below.
I am standing on the top of a cliff that is 20m above sea level (assume that I have no height; I am looking from 20m above sea level). I am looking down at a boat and the angle of depression is 35°. How far is the boat away from the cliff? Give your answer to 2 decimal places.
The first step is to create a quick sketch. The angle of depression is the difference between the horizontal and the line of vision. I am standing 20m above sea level. The horizontal will always be 20m above sea level. Therefore, I am going to create a right-angle triangle with the horizontal, line of vision and the height above sea level at the right of the diagram rather than at the left. We get the right-angle triangle that is given below.
We have been given an angle and the opposite and we are looking for the adjacent. This means that we will be using the TOA formula triangle.
We are looking for the adjacent, which means that we will be dividing the opposite by tan of the angle.
We now need to round our answer to 2 decimal places.
The number to the right of the line is a 2, which means that we round down. Therefore, the ship is 28.56m from the base of the cliff to 2 decimal places.
Example 3
This is not a word problem, but it is a slightly different trigonometry problem because we are given information that we have to ignore in order to find the unknown that we are looking for.
Calculate the height of the triangle below.
This is not a word problem, but it is a slightly different trigonometry problem because we are given information that we have to ignore in order to find the unknown that we are looking for.
Calculate the height of the triangle below.
We are looking for the height of the triangle. We can find the height of this triangle by creating a right-angled triangle from the information that we are given.
We now need to label each of the sides. We are looking for the opposite and we are given the hypotenuse and an angle. Therefore, we will be using the SOH formula triangle.
We are looking for the opposite, so we cover up O in the formula triangle, which tells us that we multiply sin of the angle by the hypotenuse.
We are rounding to 3 significant figures. When we are rounding to significant figures, we count from the left from the first non-zero number. We then place a line after the 3rd significant figure.
We now look at the number to the right of the line, which in this case is a 6. This means that we round our answer up, which is done by increasing the number that is on the left of the line by 1. The number to the left of the line is a 9, which when increased by 1 becomes a 10, which causes the 5 in the tenth position to be increased to a 6. Therefore, the answer for this question to 3 significant figures is 4.60 units, which is 4.6 units (the 0 in 4.60 units does not add anything, thus meaning that we can ignore it).