Sin, Cos, Tan

[berlin]

When I checked my edline and discovered that I had a 'D' in Trigonometry, I wasn't too flustered by the fact that I was doing poorly, but I was flustered most by the fact that I had no idea why I was pressing 'sin', or 'cos', or 'tan'... I know what they mean. I know their reciprocals. But, I don't know when to use them, and always have to look at previous problems, looking for patterns. I probably could memorize the patterns, but would that really be learning trigonometry? I still wouldn't know why I was pressing the function buttons. So, I decided to delve deep into the roots of the subject because History is a subject that I get A's in. I learned about Hipparchus, and his awful rejection of the Heliocentric model of the solar system, and his influence on Ptolemy, who, I know, had great influence on the Church. Also, I learned about cord tables, which I understood. But, I realize that tomorrow I'm going to be sitting in my Trigonometry class, still stumped over why I'm pressing the Sin, Cos, and Tan buttons. Of course, we aren't allowed to learn in trigonometry because we've got to be given tests and classwork... people don't care about learning anymore. They only care about their grade, and that's why everyone's totally fine with just memorizing things. But yeah... Can someone explain to me what a function is? In a way that I can understand... without a billion equations because that's just memorization again. And, how to the Sin, Cos, and Tan functions work? And what are they? Is it possible to draw a picture out for me? :/

[darkenlighten]

Okay so first things first: A function is anything that takes input and does something with it to give an output. For example, [math] f(x) = x^2 [/math] or stated as [math] y = x^2 [/math]. Say [math] x = 2 [/math] then your output is [math] f(x) = 4[/math] or [math] y =4 [/math], either is equivalent nomenclature.

 

So if you understand a function, we can now move to [math] Sin(\theta), Cos(\theta)[/math] and [math] Tan(\theta)[/math].

 

The best thing to remember SOH, CAH TOA, which stand for [math] Sin(\theta) = \frac{Opposite}{Hypotenuse} [/math], [math] Cos(\theta) = \frac{Adjacent}{Hypotenuse} [/math] and [math] Tan(\theta) = \frac{Opposite}{Adjacent} [/math]. You should have learned what the Hypotenuse, Opposite and Adjacent sides of a triangle are. With this you should be able to use it properly for trying to find lengths and angles.

 

Each equation is going to have 3 variables, the angle: [math] \theta [/math] and the two lengths corresponding to that angle. So you would use each one when you are given 2 out of the 3 variables and the angle is the angle residing in between the two sides.

 

We'll start there, go.

Edited September 12, 2010 by darkenlighten

[ewmon]

I was flustered most by the fact that I had no idea why I was pressing 'sin', or 'cos', or 'tan' [...] I probably could memorize the patterns, but would that really be learning trigonometry? [...] They only care about their grade, and that's why everyone's totally fine with just memorizing things. But yeah... Can someone explain to me what a function is? In a way that I can understand... without a billion equations because that's just memorization again.

Berlin, don't worry, and congratulations on rejecting the because-the-teacher-says-so attitude of modern learning.

 

Darkenlighten has it right. Sines, cosines and tangents are ratios of certain sides of a right triangle that can be computed from the value of one of the angles.

 

In short: set up a ratio with "the length you want to find" on top and "the length you already know" on the bottom. For example, you want to find the length of the adjacent side, and you know the length of the hypotenuse. So, this is Adjacent/Hypotenuse. See what Darkenlighten wrote. This is the cosine function. So, use the angle between the adjacent and the hypotenuse to compute the cosine. This is the ratio of Adjacent/Hypotenuse. Multiply this by the length of the hypotenuse, and in an odd way of looking at it, the hypotenuses will "cancel out", leaving you with the length of the adjacent after you do the math.

 

So, using values, you know the hypotenuse is 36 inches, and the angle between it and the adjacent side is 60°. You want to know the length of the adjacent side. So find cos(60°), which is ½, which means there's 1 inch of the adjacent side for 2 inches of the hypotenuse. This could also be large: 1 mile of adjacent side for 2 miles of hypotenuse. Or it could be small: 1 micron of adjacent side for 2 microns of hypotenuse. You should be measuring all lengths using the same units, so the units cancel out in the cosine's fraction, leaving a simple ratio. That's why trig functions have no units ... they''re simply ratios. So, you take the 36 inches of the hypotenuse and multiply it by ½, and you get 18 inches for the length of the adjacent side.

 

I hope this makes sense to you. Come back if it doesn't.

[berlin]

Thank you so much!! I feel extremely enlightened; when I read it all of the sudden every road block in my mind was cleared. But, then when I looked at my trig homework, I saw a problem where the angle theta and the opposite and adjacent are all unknown, and the only thing given is the hypotenuse and the alpha angle. I'm guessing that weird sign that my teacher squeezed between the opposite and hypotenuse is the alpha sign, at least... it looks a lot like a Jesus fish... .... okay, sudden religious/mathematical connection just smacked me upside the head but I don't think this is the right place to talk about theology x)

 

What do I do? D:

[darkenlighten]

Thank you so much!! I feel extremely enlightened; when I read it all of the sudden every road block in my mind was cleared. But, then when I looked at my trig homework, I saw a problem where the angle theta and the opposite and adjacent are all unknown, and the only thing given is the hypotenuse and the alpha angle. I'm guessing that weird sign that my teacher squeezed between the opposite and hypotenuse is the alpha sign, at least... it looks a lot like a Jesus fish... .... okay, sudden religious/mathematical connection just smacked me upside the head but I don't think this is the right place to talk about theology x)

 

What do I do? D:

 

I'm glad that it helped.

 

For your problem, remember that those equations don't have to be independent from each other and can all be used for the same problem. Look at the knowns and see which equation can apply in order to move on to the next part of the problem.