Find the exact value of the trigonometric expression

[QuantumCurt]

Homework Statement

Find the exact value, without using a calculator.

$$\ cos(\frac{\pi}{15}) \ cos(\frac{\pi}{45})$$

The Attempt at a Solution

I started off using a product to sum formula-

$$\frac{1}{2}[cos(\frac{4\pi}{45})+ \ cos(\frac{2\pi}{45})]$$


Now I don't know where to go. I'm trying to use a sum/difference formula, but I can't find two values from the unit circle that will add or subtract to the arguments of the cosines, so I'm thinking that's not the best way to go. I tried using a sum to product formula within the product to sum formula, but then I realized that it was just going to get me back to the original expression pretty quickly. I tried the half angle formulas, but those still don't get me to a value that's on the unit circle.

Any suggestions?

 

[verty]

Every instinct tells me this is not possible. Here is a list of some exactly known cosines but I can't see how to relate this question to any of them.

 

[QuantumCurt]

Every instinct tells me this is not possible. Here is a list of some exactly known cosines but I can't see how to relate this question to any of them.

My instinct is exactly the same. I think she entered this one onto the worksheet wrong. It's part of a review/practice for our trig final next week...and I've approached this one numerous different ways. I don't think it can be done. One of my friends that's in the same class was in our schools tutoring center yesterday, and she texted me and told me that nobody up there could figure it out either.

Anyone else have any suggestions?

 

[hilbert2]

If you can use the formulae $sin 3\theta = -4sin^{3}\theta + 3sin\theta$ and $cos 3\theta = 4cos^{3}\theta - 3cos\theta$ (which can be derived from De Moivre's formula), you can reduce this problem to knowing that $sin \frac{\Pi}{5} = \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}$ and $cos \frac{\Pi}{5} = \frac{1}{4}\left(1+\sqrt{5}\right)$. Every number of the form $sin \frac{m\pi}{n}$ or $cos \frac{m\pi}{n}$, where $m$ and $n$ are integers, can be represented exactly as an algebraic number (a root of some polynomial with integer coefficients).

 

[QuantumCurt]

If you can use the formulae $sin 3\theta = -4sin^{3}\theta + 3sin\theta$ and $cos 3\theta = 4cos^{3}\theta - 3cos\theta$ (which can be derived from De Moivre's formula), you can reduce this problem to knowing that $sin \frac{\Pi}{5} = \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}}$ and $cos \frac{\Pi}{5} = \frac{1}{4}\left(1+\sqrt{5}\right)$. Every number of the form $sin \frac{m\pi}{n}$ or $cos \frac{m\pi}{n}$, where $m$ and $n$ are integers, can be represented exactly as an algebraic number (a root of some polynomial with integer coefficients).

We barely even looked at De Moivres formula. It's a summer class, so we're cruising through a lot of it. When we looked at it, we really only looked at how to find powers of a given expression. like (sin x cos x)^n She told us that we didn't need to worry about going over any of the parts on finding roots and whatnot. I've never even seen the formulae you just posted. I'll play around with it and see what I can come up with though.

Thanks for the help.

 

[rcgldr]

I there any chance that the units are messed up and that the bottom numbers are in degrees, while the upper numbers are in radians since it's π?

 

[QuantumCurt]

I there any chance that the units are messed up and that the bottom numbers are in degrees, while the upper numbers are in radians since it's π?

Something like that is possible. She made the study guide herself, so she may have just entered something incorrectly. I got the impression that she had to hurry up and get it finished too. She was supposed to have given it to us on Monday, but she said she hadn't had any time to get it put together. We got it on Wednesday, but I don't know how much of a hurry she was in while making it.

There are a couple other questions on here that seem to possibly have a similar issue. My friend that's in the same class was having trouble with a couple other questions on it, and even the people at the schools tutoring center weren't able to figure them out...and these are people with masters degrees that tutor math everyday. I'm still trying to puzzle these other ones out though...I think I might be getting close. If I can't get them though, I may have a couple other questions to post.

Admittedly, this teacher is a bit scatterbrained and disorganized. She basically teaches right out of the book...literally going over the examples that are in the textbook in class. There have been quite a few times this semester that she's made a mistake on the board and I've had to correct her.

edit- The problem on the study guide is exactly as I posted it here though.

 

[Mark44]

It would be worthwhile to contact the instructor for clarification on this problem, to see if it is as she meant it to be.

 

[QuantumCurt]

It would be worthwhile to contact the instructor for clarification on this problem, to see if it is as she meant it to be.

That's a good idea. I think I'm going to send her an email later. I'm going to keep trying on these couple of other problems a little longer, because if I can't figure them out, I want to mention them too.

 

[lurflurf]

likely a typo
cos(pi/15) is easy cos(pi/45) is hard
homework is usually limited to multiples of pi/15 since pi/45 is a third we will need cube roots

method 1:
use the identity
$$\cos (a \, \pi)=\frac{1}{2} ((-1)^a+(-1)^{-a})$$
with a=1/45
this is easy but unsatisfying

method 2:express in square roots
cos(m*pi/n)
can be expressed in square roots when
n=2^a*3^b*5^c*17^d*257^e*65537^f
where
a=0,1,2,3,4,...
b,c,d,e,f=0 or 1
usually in beginning trig we only consider
n=2^a*3^b*5^c
45=3^2*5
the method fails
we could express it in cube roots though

method 3:we might be able to use the product to simplify for example
$$\cos \left( \frac{\pi}{9} \right) \cos \left( \frac{2\pi}{9} \right) \cos \left( \frac{4\pi}{9} \right)=\frac{1}{8}$$
I do not see any similar simplification in this problem

method 4: the largest root of
16777216 x^12-28311552 x^10-7602176 x^9+15335424 x^8+7962624 x^7-1646592 x^6-1953792 x^5-437760 x^4-17984 x^3+3744 x^2+288 x+1=0
is cos(pi/15)cos(pi/45)
yay
however the equation is difficult to solve

 

[QuantumCurt]

Wow. I think some of that is very far beyond the scope of this class. It's an introductory trig class. Our book touches on some of this stuff, like finding roots and stuff. Since it's a summer class though, we're not going to have time to go over it all. I'm going to take the few weeks between finals(next week) and the start of fall semester to go over some of that material though. The section on rotations of conics looks really interesting.

Thanks for the help!

 

[rcgldr]

I there any chance that the units are messed up and that the bottom numbers are in degrees, while the upper numbers are in radians since it's π?

If that was the case, you'd end up with 1/2 (cos(16) + cos(8)), which I doubt is what is wanted.

 

[QuantumCurt]

I just got an email back from the professor. It was indeed a typo. She said to just exclude that problem. I ended up figuring out the other few I was struggling with too. Thanks for the help. :)