Expert: Josh - 7/7/2006

Question
my question has to do with sines and cosines. i dont really understand them at all, so i was wondering if you could explain how they work. and then theres a problem that i need to know how you get the answer. i had a few people try to explain but its like right over my head

its a triangle, a right triangle , the right angle is on the bottom left corner, its labled as C

the straight above it ( the top left corner) is labled as A

and then horizontil from C is labled as angle B

the side AB is labled as c
the side CB is labled as a
the side ac is labled as b


and the question it asks :

what if angle B is 40 degrees and a = 12, what is b?

i really need to know how to do these problems, so  a lot of detail would help, cuz i cant find anyone so far who can help me. thanks

Answer
Andrea,

Sine and Cosine are functions which relate "the ratio between two sides of a triangle" to an "angle".

A function is something which takes an input value and returns an output value.

e.g., If we consider y=sin(x), this equation describes a sine function which takes an input "x" and returns an output "y". Here, x represents the angle (usually measured in degrees) and y represents the ratio between two sides of a triangle.

So, what is the difference between sine and cosine?
Ans: if we label angle ^ACB as "x" in the following diagram,

A
|
|
|
B------------C

sin(x) is defined by |AB|/|AC|, the length of the side opposite to the angle x, divided by the length of the longest side, also called the hypotenuse of the triangle; whereas cos(x) is defined by |BC|/|AC|, the length of the side adjacent to angle x, divided by the length of the hypotenuse.

A simple way to remember this is that
sine gives the ratio of "opposite"/"hypotenuse", while cosine gives the ratio of the "adjacent"/"hypotenuse".

=======================================================
Summary: The trigonometric functions relate an angle to the ratio subtended by two sides of a triangle. Specifically, if we label as shown in the above diagram,
sin(x)=|AB|/|AC| (remember, opposite/hypotenuse);
cos(x)=|BC|/|AC| (adjacent/hypotenuse);
tan(x)=sin(x)/cos(x)=|AB|/|BC| (opposite/adjacent). [#1]

Note: You need to remember the fundamental meaning of sine and cosine. Once you can picture where everything fits, you would be able to write down the correct expression, regardless of how the vertices (and sides) are labelled.
=======================================================

You did well describing the convention adopted in your question. We have

A
|
|
|
C------------B

Here, the sides AC, CB and AB have the following dimensions. Lengths: |AC|=b, |CB|=a and |AB|=c.
The angles ^ABC, ^BCA and ^CAB are denoted "B","C" and "A", respectively.

If "B"=40 degrees, a=12, to find "b", we need to relate the ratio between the sides "a","b" and the angle "B".

Using what you've just learnt, we go over the definitions of sin(B), cos(B) and tan(B).

Remember that a=|BC|, b=|AC|. We observe that |BC|=a is adjacent to angle "B" while |AC|=b is directly opposite to angle "B". Now, which trig. entity actually relates the "opposite" side to the "adjacent" side? Ans: Only tan(B) serves this purpose.

Specifically, in the context of the diagram, tan(B)=|AC|/|BC|=b/a. ...[#2]

Comment: Bear in mind, that we are equating the ratio between the lengths of the opposite and adjacent sides with respect to angle B. This is first principle. If you can master the definitions given in [#1], everything simply follows.

From [#2], we have b/a=tan(B), i.e., b=a*tan(B).
Now, let's plug in the numbers: B=40, a=12.
Therefore, b=a*tan(B)=12*tan(40). Easy to work this out on a calculator.

I hope you find this helpful. Go over what I've written. Let me know how you are finding this.

Cheers