Angle of elevation and depression

Question

i have no idea how the angles of elevation and depression work. My teacher didn’t teach us, he just gave us the problems to do, so I’m extemely confused. 

Can someone help explain these to me and hopefully walk me through a problem so I can see how to do it?

I have two examples in my book

The tallest tree in the world is reputed to be a giant redwood named Hyperion located in Redwood National Park in California, USA. At a point 41.5m from the centre of its base and on the same elevation, the angle of elevation of the top of the tree is 70°. How tall is the tree? Give your answer to 3 s.f.

 

from a 50m observation tower on the shoreline, a coastguard sights a boat in difficulty. The angle of depression of the boat is 5°. How far is the boat from the shoreline?

 

in the diagram, the height of the tower is 50m and the 5° is in the angle. They drew a line from the top of the tower, straight out across. Then another line towards the boat. 

 

Answer 1

The angles of elevation and depression are both measured from the horizontal.

 

your Eye is level with the tower, or level with the base of the tree (you’re standing in a hole, I guess). You are looking straight out (ie horizontal).

 

if you have to look up to see the top of the tree, that’s the angle of elevation.

But if you have to look down to see it, that’s the angle of depression.

 

So, in the diagram, they showed the tower at 50 m high, they showed a horizontal line .(that’s you, looking straight out), and they showed a line angling down toward the boat. The angle of depression is the angle between the horizontal line and the line to the boat.

 

how do compute the distance to the boat?

we have a right triangle, with height 50 m, and you want to know the length, because that is the distance to the boat.

 

Since this is a trig question, we’ll need an angle. The angle of depression is NOT the angle inside the triangle. Instead it is the complement of the angle in the triangle (as the diagram doubtless shows you).

 

finally, the math:

you have the angle of depression (5 degrees), the height of the tower, and you want to find the length of the triangle. Which trig function relates the adjacent to the opposite? The tangent.

 

So, tan(90-5) = x m / 50m

 

solve for x

 

the height if Hyperion requires similar thinking, but I don’t need to babble so much.

 

you have a right triangle, with length 41.5, and angle 70 degrees, and you want to discover the height. Notice this IS the angle inside the triangle, so we don’t need to subtract it from 90.

 

once again which trig function relates sdjacent to opposite? Once again, tangent.

 

So tan(70) = x / 41.5

solve for x

 

does this help?