Trigonometry - Sine, SOHCAHTOA

Solving trigonometry problems can be easy, and in this and the following posts, I will hopefully provide enough trig homework help to allow you to understand the basics.

There are 3 basic trig functions that you will encounter in your trig problems first. The first one I will discuss is SINE, and I will refer to the triangle pictured in the previous post.

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 3 functions and their ratios. It may be easier to see if you look at it with spaces: SOH CAH TOA. The SOH term is short for "Sine is Opposite over Hypotenuse."

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. 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.

Similarly, if you know some 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 sign on your calculator...)
A = 53.13 degrees

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

A quick shortcut for triangles is to understand that if you sum up the 3 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. Although, 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 to performing these basic trig functions. I hope this explains the sine function so that it is understandable. As always, please don't hesitate to comment if you're unsure or if you would like additional trig help. I'll discuss COSINE and TANGENT in the next few posts.