# Sin and Cos

How do I understand them and what they do? I've memorized "soh-cah-toa", but I still don't get what is the point of it. Like what is the application of finding something divided by another thing?

## Answers and Replies

In the context you are seeing sin and cos, trigonometry is really just a shortcut. You can use them to find various side lengths and angle measures, which is very useful in geometry. These definitions however, are the most basic.

Once you move beyond determining ratios of the sides of a right triangle, you will find that trigonometric functions can be extended to values for non-acute angles. Elementary methods usually involve the role of the unit circle. Once we can extend the functions sin and cos to the real numbers, you'll find that these functions are periodic, i.e., they repeat over and over. This is arguably their most important property. Periodic functions can be used to analyze situations involving the vibration of certain quantities or instances of heat transfer. You can often break down very complicated functions to cosines and sines.

Thanks. do you have any examples on how to find lengths and stuff? i mean if we already have the lengths, and we need to find the sin, then what is the point of sin and cos?

Mentallic
Homework Helper
Thanks. do you have any examples on how to find lengths and stuff? i mean if we already have the lengths, and we need to find the sin, then what is the point of sin and cos?
There are many physical applications, and just to name a few: http://en.wikipedia.org/wiki/Trigonometry#Applications_of_trigonometry"
But since you're only beginning the topic of trigonometry, there really aren't going to be any amazing discoveries to be had just yet.

from the sohcahtoa, you can see that $tan\theta =\frac{opp}{hyp}$
so lets say you are given a line $y=mx$ (m being the gradient), this is a line passing through the origin.
But the gradient m is calculated by $\frac{rise}{run}$ which is $tan\theta$
If you don't understand how this works, draw a line on the number plane passing through the origin (make it a positive gradient for simplicity) and then construct a line perpendicular to the x-axis touching the line. The rise is the y-value while the run is the x-value. Now $\theta$ is the angle between the x-axis and the line.

Now, you can change $y=mx$ into $y=(tan\theta) x$
And now you can find the angle that is made by the line for any positive gradient (later, this will extend onto negative gradients).

e.g. if the line makes an angle of 45o with the x-axis, then tan45o=1 so the gradient of the line is 1 (this one is common sense though).
if the line makes an angle of 60o then tan60o is $$\sqrt{3}$$ so the gradient of the line is exactly $$\sqrt{3}$$

This of course, is just a basic start to the many more deeper applications of trigonometry.

Last edited by a moderator:
How do I understand them and what they do? I've memorized "soh-cah-toa", but I still don't get what is the point of it. Like what is the application of finding something divided by another thing?
What you are missing is that it is the sine (or the co-sine) of the angle. So $$sin\theta = \frac{opp}{hyp}$$. If you look on your calculator you'll see $$sin^{-1}$$, or perhaps (not that I've ever seen it on a calculator) $$arcsin$$. What this does is it negates the sine part to get the angle. In other words, $$arcsin(sin(\theta)) = \theta$$.

However (as mentioned by snipez90) these trigonomic functions are periodic (meaning they repeat themselves). For the basic sine and cosine, it repeats every 360 degrees, or 2pi radians (but not necessarily for the others). In other words, where $$x\in Z$$ (i.e. x is an integer) $$sin(n + 360x) = sin(n)$$ (in degrees) or $$sin(n + x2\pi) = sin(n)$$ (in radians). But, if you are just using angles, this shouldn't hassle you for a while.

Trigonometric functions are used to model periodic phenomena. They are also useful in solving certain problems. Don't get discouraged, it might seem like you are learning random bits and pieces but it all comes together later on. For example differential equations required almost every piece of math I had previously been exposed to.

Let's say you have a flag pole that you want to know the height of. It isn't easy to measure the vertical height, but it is quite easy to measure a length along the ground. So, take out your measuring tape and measure off, say 20 meters from the base of the pole. From that point, you can measure the angle with a protractor by sighting the top of the pole, and you find the angle to be, oh, 35 degrees.

What do we have? The flag pole forms the opposite side of a right triangle, and the 20 meter line on the ground is the adjacent side. We know the adjacent side and we know the angle, and we are looking for the opposite, so what function deals with both opposite and adjacent? toa seems to fit. So we take the tangent of 35o, which is about 0.7. Now we have $0.7=\frac{opp}{20}$. Multiplying both sides by 20, to clear the fraction, we get $0.7(20)=20\frac{opp}{20}$. So the flag pole is about 14 meters tall.

But this is one application of right triangle trigonometry. But what if you have an obtuse triangle? soh-cah-toa doesn't apply anymore, but the trig functions still do. This is where the unit circle comes in and you will see that the circular trig functions are a bit more "natural" than the right triangle definitions.

Borek
Mentor
i mean if we already have the lengths, and we need to find the sin, then what is the point of sin and cos?
You can measure the angle, then it is a matter of just reading sin/cos value from the tables (or from your calculator) - and following calculations are pretty easy.

This is shortened down version of scenario DecayProduct described.