For this week in Math 10 I chose the concept of angle of elevation and angle of depression. I think that this is an important concept to understand because in order to find these specific angles on a right angle triangle and to know which angles they are and what you’re trying to find, you need to understand what the terms mean. This is also the most important thing that I have learned in week 4 of Math 10 because these are kinds of angles that are used in word problems for everyday scenarios often, and are therefore relevant to trigonometry that could be used in everyday life and be of use to understanding ways and situations that angles are used.
The angle of elevation in a right triangle is the angle from the horizontal upwards until it reaches an object. In an angle of elevation, the horizontal arm of the triangle is the line of sight of a person who is looking straight forwards; finding the angle of elevation determines the angle at which they would have to look upwards to see the object. With the angle of depression, it is the same idea except with an object that is below the person’s line of sight as supposed to above. For an angle of depression, the starting point is the point of the triangle that is above the object and you would find the angle from the horizontal downwards, showing the angle at which they would have to look down from their straight across line of sight in order to see the object. If you know the angle of elevation of depression, it can allow you to find the distance between the two objects, which can be very helpful and is another reason why it is an important thing to know. The angle of elevation and depression are also always the same angle.
What the angle of elevation and depression look like in a right angle triangle:
It is important to know that the angle of depression or elevation are not always one of the three angles in the right angle triangle to start with, and that you can draw another triangle to create the angle that you need to find. To find the angle of depression or elevation you first identify the starting point, which is where the angle begins and would be the higher point in an angle of depression and the lower point in an angle of elevation; the horizontal from the starting point is the first leg of the angle. The next thing you do is to determine the other point which is where the angle stops, and in many cases is the object that we are looking up or down at. We form the other leg of the angle going either upwards or downwards to this point. Finally we find the angle for the two points which will be our angle of elevation or depression.
To find an angle we use the ratio for the two sides that are involved in the problem and multiply the fraction for the ratio of the side lengths of the triangle by the ratio’s inverse function. Ex. 
In many situations there are other factors that come into play, for example when the angle doesn’t start at the ground, or we have to visualize a triangle ourselves to find the angle. The following examples show how to find an angle of depression or elevation for these situations.
Example #1: Finding an angle of depression for a specific scenario.
There is a diver diving down diagonally for 40m. If the vertical depth of the dive was 26m, what was the angle of depression of the dive?
Step 1: Draw a diagram to see how the question could be shown as a right angle triangle.
Step 2: Label the sides to determine which ratio will be used. In this question it would be the sine ratio because the sides we are using to find the angle of depression are the hypotenuse and the opposite.
Step 3: Put the side lengths of the arms of the angle we are looking for into a fraction following their ratio and find the angle by multiplying the fraction of the sides ratio by the inverse of the ratio used.
Example #2: Finding an angle of elevation for objects in my house.
Step 1: subtract the height of the cup from the height of the thermos because the angle of elevation from the top of the cup to the top of the thermos doesn’t start at the bottom of the thermos.
28cm – 8cm = 20cm
Step 2: Use the lengths of the two legs of the angle of elevation in the correct trigonometric ratio to find the angle. In this case it would be the tangent ratio because we have the side lengths for the opposite and adjacent side.
