Sin, Cos, and Tan

[Cheetah]

hopeing someone can answer this! i have the EOC's in about a week and i need help with geometry. i dont really understand how to set up sin, cos, and tan into the calculator . when you enter them it shows (for example Sin) sin(  . do i put the angle after the parenthesis and the line measure before the sin? also how would i solve the following,

the top of a tree has fallen and the top is now 10 feet from its base. it makes a 40 degree angle with the ground. how long is the fallen part of the tree

and

a ladder is put on a wall and the base of the ladder is 10 feet from the wall. the angle of elevation with the ground and the ladder is 40 degrees. how high is the wall?

Please do not just put the answer i would also like to know how to enter it into a graphing calculator. Thanks for your time!

 

[jim360]

The problem is that each calculator is different. For some calculators, you would input the size of the angle, say 45 degrees, and then hit the SIN button and it would automatically calculate this for you. In other calculators you put sin first then the numbe.r To answer this more fully I'd need to know the make of calculator.

From the looks of it, though, suppose you wanted to work out sin45 (sine of the angle of size 45 degrees), you would hit the sin button, then 45, then close brackets, then hit the equal sign. So that the display reads, " sin(45) ". For any length, this can go either before the sin - " 3*sin(45)", say - or after - "sin(45)*3", say - so long as you play it safe and include a multiplication key between the two.

Now to the problems:

Step 1: draw a diagram



(It's probably not important to be arty about it though. )

Step 2: what information do we have and what information do we want?

Here it's ADJACENT side to the known angle, and we want the HYPOTENUSE, so the answer is obtained from the cosine ratio: cosine of angle = adjacent divided by hypotenuse.

Step 3: Work out the answer.

For the next problem we go through the same steps, except by some strange coincidence it is in fact exactly the same triangle as before! This means that you have any number of different ways to answer the problem, since this time we have five pieces of information about the triangle (two sides and three angles). So, you could work it out by any of the sin, cos, tan ratios, or by Pythagoras' Theorem, or by actually drawing the triangle. (with, say, 1 foot to 1 inch), or through use of the inner product in a Euclidean Space (though this is probably just being silly).

[Ford]

alright.. Here if Mr. Magic Ford...

As you should know, you can only use SOH CAH TOA (sin, cos,tan) with a special right triangle.
A trick i learned that is quite helpful when you dont know which of the three to use is;

SOH: Sin=Opposite/Adjacent angles.
CAH: Cos= Agjacent/Hypotenuse
TOA: Tan= Opposite/Adjacent
~~~~~~~~~~~~~~~~~~~~~~
It is very easy to remember. When you have your test, just write SOH CAH TOA on the top and it will be very helpful.

Now,  you should know which angle is which.. For instance.. In you example you have

the ladder.

Well. If you get out your handy dandy notebook.. Just draw a Right angle triangle. Make the right angle the bottom left angle. Just draw a box in the angle. Now, you should have the tallest part of the triangle unlabled. (not the slanted part), the for the bottom leg of the triangle draw it out a little and put a 10 under it. then for the angle of elevation with the ground. For the angle that is also on the bottom you should label it 40 

Now, you should have 2 angles marked. one with the box (90) and one labeled 40.  Then you should have only one side labeled. (10) so you should have one open angle, and two open sides..

Well, if you know from experience, in a triangle, all sides must add up to 180 degrees. You already have the one angle at 90, so that is half. So if you wanted to, you could find the empty angle for whatever reason. just use simple math..  40+50=90 ~> 90+90 =180.

Now you know the third angle. This can be helpful for checking to make sure your answer is right later on.

Now, all you need to do is find the height of the wall. (H). Well in this case, you could use TANJIT. So when you hit the TAN button on your calculator it comes up with ~> tan(   <~ right after the ( is where you put the angle you have.

Tan could be used in either the 50 degree angle, or the 40 degree. I would use the 40 because that was givin, so you know it is right.

So, if you follow SOH CAH TOA, you will see for TOA it is tan= opposite over adjacent. Meaning... your equation would look as follows for this problem :

tan(40     H
 ------- = -------
     ~        10

in this case you see the H is DIVIDED by 10. to make the problem easier, you just move the equation..

10tan(40=h

What i did was move the dominator in the problem to the top, making it all one easy step.

if you put it in your calculator just like that, it will give you the correct answer..

H= 8.290996312ft.

Any more questions just ask..

_{BFM_»FØRЫ}

[Goalie]

Sine (abbreviated sin) takes the angle you give (in degrees or radians) and returns the ratio of the side of the triangle opposite of that angle to the hypotenuse.  i.e. Sin(30) = .5, meaning the side of the triangle opposite to that angle is exactly 1/2 of the length of the hypotenuse.
Cosine (abbreviated cos) works the same way, only it gives you the ratio of the side next to that angle to the hypotenuse.
Tangent (abbreviated tan) returns the ratio of the opposite side to the adjacent side of the angle.

The above statements may confuse you, so let's look at your problem:

I'm trying to understand your first one.  I guess another way you could put this question is that a lumberjack on a ladder sawed halfway through a tree.  The top part of the tree bent and is touching the ground.  The angle that the top part of the tree makes with the ground is 40 degrees.
Assuming that the tree was completely vertical before it was fallen, that would mean that the trunk of the tree makes a 90 degree angle with the ground, or a right angle.  That leaves the last angle at 50 degrees.  (40+50+90 = 180)

You want to find the length of the fallen part of the tree, which would be the hypotenuse.  You know all the angles, so you can do this in one of two ways:
1. 10 / cos(40).  You know the length of the side is 10 feet, and you also know the angle next to it, which is 40 degrees.  if you solve the equation cos(40)=10/h for h, you get h=10/cos(40).
2. 10 / sin(50).  You know the length of one side is 10 feet, and you also know the angle opposite to that side, which is 450 degrees.  Solving the equation sin(50)=10/h for h, you get h=10/sin(50).

A good trick to remembering how sin, cos, and tan works is the acronym SOHCAHTOA.

Sine is
Opposite over
Hypotenuse,
Cosine is
Adjacent over
Hypotenuse,
Tanget is
Opposite over
Adjacent.

(Opposite and Adjacent refer to the sides relative to the angle)

[Ford]

my post is the best!

_{BFM_»FØRЫ}

[Goalie]

Ford, your post is incorrect.  The Hypotenuse should be the longest part of the triangle, but it isn't in your post.

[jim360]

I think Ford's post is essentially correct, just with unfortunate choice of unknown. It's best to stick to h, x, y for hypotenuse and the other two sides. Also, it's "denominator" rather than "dominator". Also, what's "tanjit"?

And my post had a diagram in it.

[Goalie]

Either way, Ford gets a 0 for that question because he gave the wrong answer.  

And I've never known a calculator that requires you to put in the angle first, then press sin/cos/tan.  Of course, I haven't worked with many calculators.
Graphing calculators should let you push sin/cos/tan first, then it will give you parentheses to put in the angle.  Non graphing calculators may do it differently, but that's not what you're asking for.

[jim360]

As in, it's meant to be 8.390663612... ft

[LEGO]

Woooohoooo SOHCAHTOA!!!!   

[Cheetah]

thank you all soooooooo much! you guys have no idea what this means to me! i think i finally understand it. if you are looking for the sin of an angle and you have the hypo (my abbreviation for hypothemuse) you multiply and if you have the oposite then you divide. is this correct?
in other words using SOH CAH TOA if you are given the side of the second one (in sin the hypo) then you multiply and if you have the first one (in sin opposite) you divide.


Ps. no post was better then the other they all helped a lot!

[Ford]

 rounding...  answer is 8.3

_{BFM_»FØRЫ}

[Ford]

Ford, your post is incorrect.  The Hypotenuse should be the longest part of the triangle, but it isn't in your post.

the hypotenuse doesn't matter when you use Tanjit...

a ladder is put on a wall and the base of the ladder is 10 feet from the wall. the angle of elevation with the ground and the ladder is 40 degrees. how high is the wall?

[LEGO]

Ps. no post was better then the other they all helped a lot!

Even mine? 

Woooohoooo SOHCAHTOA!!!!   

[Cheetah]

yes lego. Especially yours