• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Exact value of sin 345.5°

  • Thread starter songoku
  • Start date
1,137
18
1. Homework Statement
Find the exact value of sin 345.5o

2. Homework Equations
Trigonometry Identities

3. The Attempt at a Solution
Don't know where to start.

Tried sin 345.5o = - sin 14.5o but stuck. Also tried multiply 345.5 with positive integer to get sin 2θ or sin 3θ or sin 4θ but also stuck

Thanks
 

Ray Vickson

Science Advisor
Homework Helper
Dearly Missed
10,705
1,720
1. Homework Statement
Find the exact value of sin 345.5o

2. Homework Equations
Trigonometry Identities

3. The Attempt at a Solution
Don't know where to start.

Tried sin 345.5o = - sin 14.5o but stuck. Also tried multiply 345.5 with positive integer to get sin 2θ or sin 3θ or sin 4θ but also stuck

Thanks
You should definitely start with 2*345.5 = 691. Now try to reduce 691 to an angle < 90 degrees, by subtracting suitable multiples of 180 or 90.
 
1,137
18
Sorry for taking long time to reply

sin 691 = sin 331 = - sin 29.

You mean finding sin 29 through the link you gave in other thread and using double angle formula?

Thanks
 
1,897
200
This is impossible. You can't get an exact value of the sign for any angle in degrees that's not divisible by 3. You won't ever get rid of the factor 3 by halving/doubling adding or subtracting angles.
 

SteamKing

Staff Emeritus
Science Advisor
Homework Helper
12,794
1,665
This is impossible. You can't get an exact value of the sign for any angle in degrees that's not divisible by 3. You won't ever get rid of the factor 3 by halving/doubling adding or subtracting angles.
I think you mean sine. The trig function is called the 'sine'.
 

Ray Vickson

Science Advisor
Homework Helper
Dearly Missed
10,705
1,720
This is impossible. You can't get an exact value of the sign for any angle in degrees that's not divisible by 3. You won't ever get rid of the factor 3 by halving/doubling adding or subtracting angles.
No, that is incorrect. A paper giving exact algebraic formulas for the sine of all angles from 1 to 90 degrees, in 1 degree increments, has been published on-line (with proofs included). It was done as a retirement project by an ex-professor of mathematics; for a precise citation, see one of my responses in the previous thread by user 'songoku' on a related topic.

Note added in edit: I see that BvU has already dealt with this issue, in a post that appeared on my screen only after I pressed the 'enter' key.
 

haruspex

Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
31,480
4,623
To go further, I believe it should be possible to find an exact representation of cos(pi/n) only involving square roots if and only if there exists a ruler and compass construction for a regular n-sided polygon. As is well known, that is possible whenever n is the product of a power of 2 and distinct Fermat primes. That is enough to get all multiples of 1 degree as well as pi/17 etc.

But if we allow other surds then there are more possibilities. Since cos(nx) can be expanded as an nth order polynomial in cos(x), and cos(x)=-cos(pi-x), we can expand cos(4pi/7)=-cos(3pi/7) to obtain a quartic in cos(pi/7).
I feel there should be a generalization of the Fermat primes that corresponds to roots up to cubic and quartic, but I'm not aware of such.
 

SammyS

Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,159
915
°
To go further, I believe it should be possible to find an exact representation of cos(pi/n) only involving square roots if and only if there exists a ruler and compass construction for a regular n-sided polygon. As is well known, that is possible whenever n is the product of a power of 2 and distinct Fermat primes. That is enough to get all multiples of 1 degree as well as pi/17 etc.
Doesn't that only get you down to multiples of 3° ?
 

haruspex

Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
31,480
4,623
°

Doesn't that only get you down to multiples of 3° ?
Sorry, yes, 3°. To get to 1° you need to use the cos(3x) expansion or similar, so does involve cube roots.
 

Want to reply to this thread?

"Exact value of sin 345.5°" You must log in or register to reply here.

Related Threads for: Exact value of sin 345.5°

  • Posted
Replies
5
Views
12K
  • Posted
Replies
3
Views
3K
  • Posted
Replies
10
Views
18K
  • Posted
Replies
24
Views
2K
Replies
2
Views
7K
Replies
4
Views
17K
  • Posted
Replies
3
Views
4K
Replies
6
Views
3K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top