so this is about triangles.. sooo the following is a bit of helpful reminders that I keep on my computer to help me remember how to fit the trig functions to triangles.. I strongly suggest you copy it and keep it where you can look at it often.
Use SOH CAH TOA to recall how the trig functions fit on a triangle
SOH: Sin(Ф)= Opp / Hyp
CAH: Cos(Ф)= Adj / Hyp
TOA: Tan(Ф) = Opp / Adj
I use this anytime I run into triangles or need some help with sin or cos
now the problem , 3 ladder 10, 12, & 15 feet. Alex wants to get to 8 feet.
the problems is also telling you that you can use t..... and then , the words are cut off.. but I know they were going to say Tan ... next.. :P
b/c Tan is how you figure out problems with the adjacent side and the opposite side. like this problem. Look at TOA above. use that to recall how the parts fit in the formula
Tan(∅) = Opp / Adj
they give us the Opp side of 8 feet in the problem
then they also tell us the Hyp of the triangle which is each of the ladders length. Then they ask us what is the Adj sides length?
So we also need to solve the triangle with the know hyp (ladder length).. uggg, this problem is long. Then we can solve the dist. from the wall or Adj side length.
it's two steps, if you want to think of it that way. You're supposed to be pretty confident with trig functions. I'm guessing this is a trig class.. right?
let's solve for the 3 different angles that the ladders make , each going to 8 feel. Obviously, nobody would really do this with a ladder they would just lean it against the wall . and if it's taller than where they want to climb, they would just go up part way. so anyway, find the 3 different angles.
look above to see which formula to use.
I like SOH b/c it seems to have all the pieces of the triangle we want to work with.
ladder 1 ( 10')
Sin(∅) = Opp / Hyp
Sin(∅) = 8 / 10
∅ = arcSin (4/5)
[ first, yes, I just reduced the fraction, then I did the arcSin on both sides, I think you might know how to do that already ? ]
∅ = 53.13010 °
( yes, I used my calculator to find that, calculators are okay to use when figuring out non standard angles )
ladder 2 (12')
Sin(∅) = 8/12
∅ = arcSin (2/3)
∅ = 41.81031°
ladder 3 (15')
Sin(∅) = 8/15
∅ = arcSin (8/15)
∅ = 32.230952°
now use our Tan function to find the Adjacent side which is the distance from the wall
Tan(∅)= Opp / Adj
Adj = Opp / Tan(∅)
( I did some quick algebra to move the side we want to solve for, now plug and chug all 3 angles )
Adj = 8 / Tan(53.13010)
Adj = 6.0000005
Adj = 8 / Tan(41.81031)
Adj = 8.94427
Adj = 8 / Tan(32.230952)
Adj = 12.688577
so the 10' ladder is 6 feet from the wall
the 12' ladder is 8.9 feet from the wall
the 15 foot ladder is 12.7 feet from the wall.
I really don't think that 15' ladder is going to stay on the wall.. if Alex climbs it... it's way way too far out... it will just fall straight down the wall :/ Maybe another math problem for the forces involved :P