Friday, April 27, 2007

Trigonometry - Cosine Law, Theorem of Pythagoras

The Cosine Law works similarly to the Sine Law that I have already discussed. The Cosine Law is the general form of the Pythagorean Theorem, which itself applies strictly to right angle triangles. Therefore, this law allows us to work with any triangle. It's a bit more of an equation to remember than the Sine Law unfortunately, but it is extremely useful. Here is the equation:

c(squared) = a(squared) + b(squared) -2abCosC

(I don't see a way to type superscript text on blogger, unfortunately.)

So, as you can see, the Cosine Law is useful for finding the third side of a triangle when any 2 sides and the angle between them is known. Let's try an example:

So by the Cosine Law:
c^2 = 6^2 + 8^2 - 2(6)(8)Cos(60)
c^2 = 36 + 64 - 96Cos(60)
c^2 = 36 + 64 - 48
c^2 = 52
c = 7.2

Now that you have done that, you have obtained a complete ratio to use with the Sine Law! And now, you can find the rest of the angles to fully describe the triangle!

Sin(60)/7.2 = SinB/8
Sin(60)/7.2 = SinA/6

I'll leave those for you to solve. But that's it! Using a combination of the Cosine Law and Sine Law, you can completely solve any triangle that you are given. They are extremely powerful and useful equations!

Also, on a side note... as I mentioned at the start of this post, the Cosine Law is a generalization of the Pythagorean Theorem, which specifically applies to right angle triangles. You can see that if you are working with a right angle triangle and substitute in 90 degrees to the Cosine Law, it reduces down to the Pythagorean theorem:

c^2 = a^2 + b^2 -2abCosC
Cos(90) = 0... so the 2abCos(90) term reduces to 0
and so
c^2 = a^2 + b^2... The Pythagorean Theroem! :)

Trigonometry - Sine Law (Law of Sines)

The trig functions that I've discussed so far (Sine, Cosine, and Tangent) will be incredibly useful to you when working specifically with right angle triangles. However, of course, not all triangles have a 90 degree angle in them. So can you still use these functions? Well, yes, but in a different way. One way is through application of the Law of Sines.

Let's consider a triangle that has three different angles, none of which are right angles. The standard naming scheme still applies, although now there is no hypotenuse (remember, the hypotenuse is opposite to the 90 degree angle).

But what is sine law?  The trigonometry sine law says that the ratio between the sine of an angle and the side opposite to it will be equal for all three angles. In other words:

a/SinA = b/SinB = c/SinC

Since you only really work with 2 ratios at a time, with a little rearranging, you can see that:

aSinB = bSinA

So let's try some sine law examples. Say we have a triangle with 1 known angle of 40 degrees, sides 4 and 6 units long. Find angle B:

The Sine Law says that the ratios of angle to opposite sides will be equal. With this, we know how to find an angle using sine law.  So we have:

6 x (Sin 40) = 4 x (SinB)
SinB = 0.964
B = 74.62 degrees

Now, if the question had asked to find all the angles in this triangle, you can simply say that since the 3 angles in the triangle will add up to 180, you can just subtract your known angles from 180 to get the third angle:

180 = 40 + 74.62 + C
C = 65.38

And now that you have angle C, you can use the Law of Sines with that angle to solve for the final unknown side:

cSinA = aSinC
c x (Sin 40) = 4 x (Sin 65.38)
c = 5.66

Whenever you are given a triangle that does not include a right angle, but you are provided with 1) an angle, 2) the length of the side opposite to that angle, and 3) any one of either the other sides or angles, then you can use the Sine Law.  Now you know how to find an angle using sine law!

Tuesday, April 24, 2007

Trigonometry - Tangent, SOHCAHTOA

So far, I've explained the concepts of Sine and Cosine... the third basic trigonometry function is TANGENT. If you've been following along and understanding those lessons, tangent isn't anything new for you. The tangent function of a right triangle relates an angle of the triangle to the ratio of its opposite side and its adjacent side. It does not refer to the hypotenuse at all. Referring to the same triangle we've been looking at, you can therefore see that:

TanB = b/a

In general terms, TanB = opposite/adjacent. We write "Tan" as the shorthand form for tangent.

Naturally, SOH CAH TOA helps us remember tangent as well. As you can probably guess by now, the "TOA" component stands for "Tangent is Opposite over Adjacent."

Again, try it out for yourself to see how all these functions work. See that in this triangle:

TanB = 3/4, so...
B = 36.87 degrees

These are the basic definitions for the standard trig functions, and they all work the same way to provide you with the same measurements for the angles or sides... the only things that are different are the sides that are being included in the ratios! So, pay attention to what sides you are looking at!

Monday, April 23, 2007

Trigonometry - Cosine, SOHCAHTOA

In continuing with my trig homework help series, this post will explain another of the basic trig functions, the cosine function.

Solving trigonometry problems that use the cosine function is extremely similar to the sine function explained in the last post. The cosine function relates an angle of a right angle triangle to the ratio of its adjacent side and the hypotenuse. The ADJACENT side to the angle is one of the sides that makes up the angle, but is not the HYPOTENUSE (the longest side of the triangle). Referring back to the same trig triangle we've been working with, it can be seen that side "a" is adjacent to angle B, and so:

CosB = a/c

In more general terms, CosB = adjacent/hypotenuse. We write "Cos" as the shorthand form of Cosine.

SOH CAH TOA helps us to remember the cosine function as well. The "CAH" term is short for "Cosine means Adjacent over Hypotenuse."

Take a look at the previous labeled trig triangle and see for yourself how the cosine function works. If you can do the sine function without problems, cosine shouldn't give you any difficulties either in your trig homework. Once again, it is absolutely important to understand the relative namings of the sides.

Sunday, April 22, 2007

Trigonometry - The Sine Function and SOHCAHTOA Explained

Solving trigonometry problems can be easy, but it first requires you to have a solid understanding of the basic trig functions.  There are three of these functions, and the first one I will discuss is SINE.  I will refer to the triangle pictured in the previous post.  In subsequent posts, I will highlight the other two functions: cosine and tangent.

The sine function relates an angle of a triangle to the ratio of its opposite side and the hypotenuse. So, if we look at angle B (and keeping in mind the notation for sides), we can see that:

SinB = b/c

In more general terms, SinB = opposite/hypotenuse. We write "Sin" as the shorthand form of Sine.

Similarly, SinA = a/c (again, it is opposite/hypotenuse... remember how I said it was important to understand the RELATIVE notation scheme!).

SOHCAHTOA is the trig acronym that describes the three basic functions and their ratios. It may be easier to see if you look at it with spaces: SOH CAH TOA. The one that we are interested in this post is the SOH term, which is short for "Sine is Opposite over Hypotenuse."  There are many other mnemonics that can be used to remember the order of these relationships, as you can see by the multiple comments below.  Thanks to everyone who submitted something!  As you can see, there is no one right way to have soh cah toa explained.  Whatever works for you to help you remember is all that is important.

Working with the Sine function is fairly straightforward, and usually just a matter of plugging the appropriate numbers into the ratio.

For example, you can imagine a triangle with known side lengths, and be asked to find the angles. This is a very common and simple question that you will get.  In this case, you would say Sin(B) = opposite/hypotenuse (where you substitute in the known values for the sides.) This will give you Sin(B) = "some value." And now, just as in working with addition or multiplication, when you want to solve for a specific variable, you have to isolate it... and to isolate it, you must do the same thing to both sides. Therefore, to get rid of the Sine, you must do 'inverse sine' to each side (which is usually the same calculator button, but pushing SHIFT to access it). Then you'll get B = inverse sine of (some value), which is your answer.  Technically, you are taking the arcsine of the ratio, though I will cover than in a future post in more detail.

Similarly, if you know a mixtures of some of the sides and some angles, you may be asked to find the unknowns. You can then say Sin(known angle) = opposite/hypotenuse (where one of these sides is known and the other unknown), and then just simply solve for the unknown side length.

Here's a quick triangle example:

So from this triangle, we can tell that:
SinA = (4/5)
SinA = 0.8 (now push inverse sin on your calculator...)
A = 53.13 degrees

SinB = (3/5)
SinB = 0.6
B = 36.87 degrees

A useful trick for quickly solving triangles is to understand that if you sum up the three angles, they will always total 180 degrees. So for this triangle, after solving angle A, you could subtract it and 90 from 180 to find B. Of course, this defeats the purpose of practicing Sine in this example... ;)

On the other hand, if we already knew that angle B = 36.87 degrees, and we wanted the length of the unknown hypotenuse, then we do:
Sin(36.87 degrees) = 3/hypotenuse
0.6 = 3/hypotenuse
hypotenuse = 3/0.6 = 5

As you can see, there really isn't anything complicated about performing these steps. For the most part, it's simple arithmetic with a few things about triangles thrown in for good measure.  I hope this explains the sine function so that it is understandable, and I also hope that now that you have had sohcahtoa explained, that makes more sense too. As always, please don't hesitate to comment if you're unsure or if you would like additional help. I'll discuss cosine and tangent in the next few posts.

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