Question

Emir is standing in a treehouse and looking down at a swingset in the yard next door. The angle of depression from Emir's eyeline to the swingset is 33.69°, and Emir is 10 feet from the ground. How many feet is the base of the tree from the swingset? Round your answer to the nearest foot.

Answer 1

Hey there! Can you draw a figure for this?

Do you know which trignometric function that we'll need to use? sin, cos, or tan?

Answer 2

@TheSmartOne there is no figure to be seen and it doesn't specify which trigonometric function to use and that is why i am very confused

Answer 3

Keep this in mind about the angle of depression
https://www.mathsisfun.com/geometry/images/angle-elevation.svg

Answer 4

15 feet
18 feet
20 feet
24 feet
these are the answer choices and ok

Answer 5

ill take a look

Answer 6

I know there is no image given but I wanted to see if you'd be able to start by drawing one

Look at the image I just shared. So we know that's the angle of depression, and Emir is 10 feet in the air
And we're trying to calculate the distance from the base of the tree to the swing set

Answer 7

Once we can draw what's being explained, then we can try to figure out if we need to use sin, cos, or tan

Answer 8

okay, I semi- understand. Can you tell me how we would start attacking this problem?

Answer 9

we need to draw an image
trignometry is all about triangles

So we have Emir and he's looking at the swing set
And then you have the tree which Emir is on, 10 feet in the air

and then you have the distance from the tree to the swing set

Do you see the triangle?

Answer 10

I think so i will attempt to draw it

Answer 11

that could be totally wrong though... i dont knoww and then the bottom part is the ground

Answer 12

The distance should be along the base

but you're getting the idea
https://imgur.com/a/TFWF4VB

So since you know the angle of depression, can you find the angle x that is part of the triangle?

Answer 13

and then do you know
soh cah toa

which one would we need to use?

Answer 14

Since we are focused on the adjacent and the opposite would we use tangent?

Answer 15

Spot on!

Now we just need to find angle x, and we already have the adjacent side and so we'll be able to solve for the distance between the tree and the swing set

Answer 16

would we have to plug it in into the calculator?

Answer 17

to calculate x? uhh I mean you have to know what to plug in if you don't want to do the math manually

but look at it

what would x and \(\theta\) add up to?

It makes a right triangle so it adds up to 90. So we know one value, so all you have to do is subtract that from 90 to find the value of the other

Answer 18

I should have probably made the distance between the tree and swing set as x but yeah just don't get confused
x here is the angle

The distance which is what the question wants us to find is just the distance and we can call that 'd' if you want
These are all variables and we can use whichever letter we want as long as we define what it is in the beginning

Answer 19

but where would we subtract the 90 from since both angles are unknown

Answer 20

your question gives you the angle of depression...

Answer 21

oh oh ohhhhhh so we would do 90-33.69

Answer 22

Refer to the original image I shared

\(\color{#0cbb34}{\text{Originally Posted by}}\) @TheSmartOne
Keep this in mind about the angle of depression
https://www.mathsisfun.com/geometry/images/angle-elevation.svg
\(\color{#0cbb34}{\text{End of Quote}}\)

Answer 23

Yup!

Answer 24

which is 56.31

Answer 25

as our last angle

Answer 26

And then you correctly said we need to use tan

So can you set it up?

now that we know the angle, all we have is the missing length of the opposite side, which is the distance from the tree to the swing set

Answer 27

tan= opposite/adjacent
so it would be tan= x/10

Answer 28

yes but don't forget the angle

Answer 29

tan(56.31)= x/10

Answer 30

we always say tan = opposite/adjacent

Because it's easier to say

But it's important to not forget that it's all relative to the angle

Answer 31

perfect! Now solve for x, which is our distance

Answer 32

And our question asks us to round to the nearest foot

Answer 33

could you do that on the calculator on this website?

Answer 34

uh no because it's not a scientific calculator so it doesn't have the ability to do tan(56.31)

Answer 35

You also have to make sure you're doing degrees and not radian

https://www.google.com/search?&q=tan%2856.31+degrees%29

Answer 36

so rounded it would be 15 feet

Answer 37

And that's your answer!

Answer 38

wow i never thought i would understand this. Thank you!!!

Answer 39

It was my pleasure!! :)

Answer 40

And welcome to QuestionCove!!

Answer 41

Thank you so much!

Answer 42

Don't mention it 😊