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 Trigonometry Trigonometry Math Forum

 July 6th, 2017, 02:16 AM #21 Newbie   Joined: Jul 2017 From: Somewhere Posts: 13 Thanks: 0 A multiplication symbol  July 6th, 2017, 02:16 AM #22 Newbie   Joined: Jul 2017 From: Somewhere Posts: 13 Thanks: 0 I have no idea how to use the math symbols here since I'm pretty new to the forums. Last edited by skipjack; July 6th, 2017 at 05:56 AM. July 6th, 2017, 06:17 AM #23 Global Moderator   Joined: Dec 2006 Posts: 20,916 Thanks: 2199 The notation sin θ means sin(θ), i.e., the sine function of θ, not the product of sin and θ. Note that θ may be in radians or degrees. There is a fixed ratio between the two, 180$^\circ\!$ (180 degrees) being equal to $\pi$ radians. July 6th, 2017, 07:24 AM   #24
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Quote:
 Originally Posted by Sigmarus I have no idea how to use the math symbols here since I'm pretty new to the forums.
No worries!

Mathematics notation makes use of all sorts of letters and symbols and it can be very confusing what they mean sometimes.

Whenever you see letters being used, you might find that they're being used in a different way than just algebraic letters.

What you need to understand specifically here is functions.

----- functions -----

In simple terms, functions are a little bit like a recipe. They:

1. take some input (ingredients);
2. do something to that input to make something new (cooking); then
3. spit the result back out (cake!).

Therefore, let's make a function and call it $\displaystyle f$. This is a common letter used for functions. The trigonometry functions you are using are the same, but they have different names and are written down as $\displaystyle \sin, \cos$ and $\displaystyle \tan$ instead. Most mathematicians use single letters for functions like $\displaystyle f$ and $\displaystyle g$, but not always...

Let's make the function take any single number as input. Let's call that number $\displaystyle x$.

What's the function going to do? Well, let's make it do something simple, like add 4 to whatever the number is.

The notation for writing out this particular function is then

$\displaystyle f(x) = x + 4$

So.... you specify the function name, $\displaystyle f$, then write next to it the inputs in brackets, $\displaystyle f(x)$, then you put what the function does on the other side of the equation.

This is not f multiplied by x.

so what would our function do if our input number, x, is 3? We can put it into the function like this:

$\displaystyle f(3) = (3) + 4 = 7$

so we've got our result, which is 7. This has got a name... it's called 'substitution' (like in football).

Let's try again with a different number, say 120...

$\displaystyle f(120) = (120) + 4 = 124$

We get a new answer, 124. Functions give different answers depending on what the input is. Therefore, they are used to describe processes, algorithms or ways of doing things.

----- algebra with functions -----

Think of the function name, in this case $\displaystyle f$, as being 'glued' to the $\displaystyle (x)$ for the purposes of algebra. You can then treat the whole thing, $\displaystyle f(x)$, as if it were a single algebraic letter!

For example... subtract 4
$\displaystyle f(x) - 4$

Multiply by 2
$\displaystyle 2(f(x) - 4)$

Expand brackets:
$\displaystyle 2 f(x) - 8$

See? $\displaystyle f(x)$ behaves like a single algebraic letter.

$\displaystyle \tan(x)$ is similar... it's a function where you can consider the 'x' as being input for the 'tan' function. You should think of the 'tan' and the 'x' as being glued together. You can do regular algebra with the whole thing.

Note: just for the trigonometry functions like $\displaystyle \sin, \cos$ and $\displaystyle \tan$, most people skip writing the brackets in, so they just write down $\displaystyle \tan x$, but they do mean $\displaystyle \tan(x)$ Quote:
 But what's the difference between tan(34) and tan x 34 :/ since tan is a function? So if atan(b)=asin(c) It can be tan(b)=sin(c) as a is not connected to the function?
tan(34$\displaystyle ^\circ$) is the tan function applied to an angle of 34 degrees

tan(34$\displaystyle ^\circ x$) is the tan function applied to an angle of 34 degrees multiplied by x

34 tan x is the tan function applied to x, then multiplied by 34.

tan x 34 is confusing and weird... the person needs to write it down in one of the three ways above instead. You certainly can't multiply a 'tan' and a '34'... the 'tan' is a function and it needs an input to make sense.

Note on inverse functions:

You've noticed that there's a function called arctan x. This is the name for the inverse function of tan x. It basically does the opposite thing as tan.

tan does the following:

angle -> function -> number

but arctan does this:

angle <- function <- number

so if you calculate
$\displaystyle \tan (45^{\circ})$, you get 1.

If you calculate
$\displaystyle \arctan (1)$, you will get $\displaystyle 45^{\circ}$... same thing, but the other way round.

The tan function takes the angle as input and spits out a number as output.
The arctan function takes the number as input and spits out an angle as output.

Some people shorten the 'arc' to 'a' so you get asin, acos and atan. I don't like this personally because it can make things confusing, but it's very common and you should watch out for it.

You cannot just do:

$\displaystyle \text{atan} x = \text{asin} y$

Divide a on both sides:
$\displaystyle \tan x = \sin y$

because the 'a' is part of the name of the function... it's not an algebraic letter. If you actually divided both sides by a, you would get

$\displaystyle \frac{\text{atan} x}{a} = \frac{\text{asin} y}{a}$

which is not very interesting to be honest!

Anyways.... hope this helps. Last edited by skipjack; July 8th, 2017 at 08:00 PM. July 8th, 2017, 03:55 AM #25 Newbie   Joined: Jul 2017 From: Somewhere Posts: 13 Thanks: 0 Thanks! However, I don't quite understand the last part; do you mind explaining a bit more? Thank you! Last edited by skipjack; July 8th, 2017 at 08:09 PM. July 8th, 2017, 01:18 PM   #26
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Quote:
 Originally Posted by Sigmarus Recently, I have just learnt a bit about the basics of trigonometry and have some questions about it. Such as why is it when 2cos(θ)=√2, it will be cos$^{-1}$(√2 /2) Thanks.
A lot of helpful responses here, but I would love to add something else...Please take it with a grain of salt.

Taking a step back from everything, understanding sin(x), cos(x), tan(x), cot(x), etc. as relationships to sides of a triangle will help tremendously. Also having a general understanding of the unit circle and how a circle relates to a triangle would be helpful.

For example, if we have a right triangle with an angle = (x) , sin(x) = opposite/hypotenuse. Essentially...

sin(angle) = [(opposite leg of triangle from angle) / (hypotenuse of triangle)].

And we can play with this just like any other algebraic function.

arcsin(sin(x)) = arcsin[(opposite)/(hypotenuse)]

The left side will just be x after both of these operations. They will cancel each other. Similar to how raising a square root to the power of 2 will cancel the root.

=> x = arcsin[(opposite)/(hypotenuse)]

Now given the length of the opposite leg and hypotenuse, we have a ratio. Taking the arcsin of this ratio will reveal the desired angle to us.

Once the geometrical relationship is understood, I believe working with sine, cosine, etc. functions will be a lot more straight forward.

Jacob

Last edited by skipjack; July 8th, 2017 at 07:49 PM. July 8th, 2017, 08:10 PM   #27
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Quote:
 Originally Posted by Sigmarus I don't quite understand the last part
Which particular line was difficult to understand? July 9th, 2017, 04:33 AM #28 Newbie   Joined: Jul 2017 From: Somewhere Posts: 13 Thanks: 0 The arctan part :/ Tags basic, questions Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post awholenumber Algebra 2 June 26th, 2017 02:04 AM football Physics 2 June 8th, 2010 08:07 AM aneuryzma Calculus 2 September 13th, 2009 03:40 AM Amir Kiani Applied Math 4 September 7th, 2009 12:56 PM Amir Kiani Calculus 4 July 31st, 2009 05:44 AM

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