Uses of sinx^2 + cosx^2 = 1 watch

Thorium · 09-01-2015 11:54

Hi Guys

I'm currently doing the Extended Project on various proofs/theorems and their applications. Pythagoras' theorem is one of the theorems I am writing about and I really want to talk about sinx^2 + cosx^2 = 1 because it's used a lot in the A level mathematics course, but I don't actually know what it is used for. You can use it to rearrange equations and such to solve them but what is an example of when this can practically be applied?

Any suggestions would be great!

rayquaza17 · 09-01-2015 21:44

I think sin^2(x)+cos^2(x)=1 is Pythagoras' theorem?

I don't think you can say any specific uses it has - that's like saying what are the uses of y=3x or x=2.

Mr M · 09-01-2015 21:49

(Original post by Thorium)
Hi Guys

I'm currently doing the Extended Project on various proofs/theorems and their applications. Pythagoras' theorem is one of the theorems I am writing about and I really want to talk about sinx^2 + cosx^2 = 1 because it's used a lot in the A level mathematics course, but I don't actually know what it is used for. You can use it to rearrange equations and such to solve them but what is an example of when this can practically be applied?

Any suggestions would be great!

This is Pythagoras' Theorem in a trigonometric form.

atsruser · 09-01-2015 22:29

(Original post by Thorium)
Hi Guys

I'm currently doing the Extended Project on various proofs/theorems and their applications. Pythagoras' theorem is one of the theorems I am writing about and I really want to talk about sinx^2 + cosx^2 = 1 because it's used a lot in the A level mathematics course, but I don't actually know what it is used for. You can use it to rearrange equations and such to solve them but what is an example of when this can practically be applied?

Any suggestions would be great!

I'm not sure what you want exactly but maybe you could mention that if you derive this from a right angled triangle with angles less than 90 degrees that it:

1. implies that the coords of a point on the unit circle in the first quadrant of the x-y plane are $\cos\theta, \sin\theta$ (why?)

2. implies that $|\sin\theta| \le 1, |\cos\theta| \le 1$ (why?)

3. both of which imply that you can extend the meanings of $\sin\theta, \cos\theta$ for $\theta > 90$ (how?)

4. from which you can imply a bunch of identities for sin and cos

5. and you can imply further that sin and cos are periodic (why? what's the period?)

Gaiaphage · 09-01-2015 22:39

It is used a huge amount in C3 and C4 trigonometry - the syllabus can be found here and it should explain what it's used for.

One example is to answer a very common question such as

"Express 3 cos x° + sin x° in the form R cos (x − α)° where R > 0 and 0 < α < 90"

where it is used to find R.

If you're googling the uses, you may also want to google the formulae tan²x + 1 = sec²x and cot²x + 1 = cosec²x as they're the same formula rearranged but also used at a higher level.

omegaSQU4RED · 09-01-2015 22:51

I can think of examples where I've seen the theorem being used, but that probably wouldn't be very interesting as it might just come up in some general trigonometric computations. If you're doing an EPQ, I would try and suggest maybe going outside of the A-level maths syllabus a bit - it'll help your general interest in the subject more as well.

If you want a good topic about proofs - I suggest reading up on propositional logic and how it relates to different methods of mathematical proof (e.g. proof by contrapositive, proof by contradiction).

Gaiaphage · 10-01-2015 01:47

Also - remember your EPQ is not allowed to overlap with your A level syllabus, you may need to rethink this a bit

rayquaza17 · 10-01-2015 07:07

(Original post by Gaiaphage)
Also - remember your EPQ is not allowed to overlap with your A level syllabus, you may need to rethink this a bit

??
Yes it is.

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Thorium · 10-01-2015 13:50

(Original post by Gaiaphage)
It is used a huge amount in C3 and C4 trigonometry - the syllabus can be found here and it should explain what it's used for.

One example is to answer a very common question such as

"Express 3 cos x° + sin x° in the form R cos (x − α)° where R > 0 and 0 < α < 90"

where it is used to find R.

I'm doing my A levels now and I know that you can use it to rearrange equations and get it into just one function i.e. in terms of sin or in terms of cos, but I don't know why this is useful. Why do we need to solve for x when using trigonometric functions?

Stickman · 10-01-2015 13:56

A lot of identities like those are used in Engineering. For example, when given a signal you need to be able to convert the function into different forms and you need those trig identities for manipulation. As a poster above has said, it's more needed in computation than actual end use.

tombayes · 10-01-2015 15:24

(Original post by atsruser)

2. implies that $|\sin\theta| \le 1, |\cos\theta| \le 1$ (why?)

really? - how?

atsruser · 10-01-2015 15:41

(Original post by tombayes)
really? - how?

We have $\sin^2\theta = 1 - \cos^2\theta \le 1$ which is satisfied by $-1 \le \sin \theta \le 1$ and similarly for cos.

This suggests that we can start thinking about sines represented by distances on the -ve x-axis.

tombayes · 10-01-2015 17:55

(Original post by atsruser)
We have $\sin^2\theta = 1 - \cos^2\theta \le 1$ which is satisfied by $-1 \le \sin \theta \le 1$ and similarly for cos.

This suggests that we can start thinking about sines represented by distances on the -ve x-axis.

well assuming the series definitions of sin(x) & cos(x)

sin(2i)^2+cos(2i)^2=1
|sin(2i)|>1

Gaiaphage · 10-01-2015 19:21

(Original post by rayquaza17)
??
Yes it is.

No it's not (AQA at least), your supervisor has to sign to say that there is no overlap with your A level subjects before you can submit your project.

rayquaza17 · 10-01-2015 19:32

(Original post by Gaiaphage)
No it's not (AQA at least), your supervisor has to sign to say that there is no overlap with your A level subjects before you can submit your project.

Where exactly does it say this?

Impossible to do a maths EPQ without overlap.

I did an EPQ and I included a lot of the stuff I leaned at A Level and I got nearly full marks on my EPQ.

MathMeister · 10-01-2015 19:48

(Original post by rayquaza17)

I did an EPQ and I included a lot of the stuff I leaned at A Level and I got nearly full marks on my EPQ.

Was that about logarithms? (I read on somebody's profile but forgot who)
I wonder how much you could really put about them hehe

atsruser · 10-01-2015 20:03

(Original post by tombayes)
well assuming the series definitions of sin(x) & cos(x)

sin(2i)^2+cos(2i)^2=1
|sin(2i)|>1

I think that it's pretty clear that we're not talking about functions of a complex variable here.

rayquaza17 · 10-01-2015 20:03

(Original post by MathMeister)
Was that about logarithms? (I read on somebody's profile but forgot who)
I wonder how much you could really put about them hehe

Yeah I did it about logarithms.
I wrote about their history, explained some of the maths behind them, and showed how they can be used in real life. I also made up some questions and solutions to them as well.

The actual maths wasn't that much beyond a-level tbh. But I don't think that mattered because my project was a website that was meant to help a-level students. Not sure how I could have achieved that without overlapping with a-level maths!

I took most of my education information off my profile because I wasn't sure if people actually read it lol. I guess people do!

Mr M · 10-01-2015 20:49

(Original post by rayquaza17)
Yeah I did it about logarithms.
I wrote about their history, explained some of the maths behind them, and showed how they can be used in real life. I also made up some questions and solutions to them as well.

The actual maths wasn't that much beyond a-level tbh. But I don't think that mattered because my project was a website that was meant to help a-level students. Not sure how I could have achieved that without overlapping with a-level maths!

I took most of my education information off my profile because I wasn't sure if people actually read it lol. I guess people do!

Does the website still exist? I want to read it now.

TheBBQ · 10-01-2015 20:54

It is used often when evaluating integrals.

For example finding the derivatives of the arcsin, arccos, arctan.

And as we know, calculus is extremely applicable.

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