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Trigonometry Find angles when circumference is divided into 5 unequal parts

Ragnarok

Member
Jul 10, 2013
41
Hello, I am using a very old textbook from 1895, Loney's Trigonometry, which poses the following problem:

If the circumference of a circle be divided into 5 parts, which are in A.P., and if the greatest part be 6 times the least, find in radians the magnitudes of the angles that the parts subtend at the centre of the circle.

The answers are given as \(\displaystyle \frac{4\pi}{35},\frac{9\pi}{35},\frac{14\pi}{35}, \frac{19\pi}{35},\frac{24\pi}{35}\) radians.

The trouble is that I have no idea what is meant by "5 parts, which are in A.P.". There is no diagram and the only reference to points A and P I can find is in a much earlier diagram, where AP is an arc subtending an angle of one radian. Does anyone have any ideas, based on the specifications of the problem? Thank you!
 

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
Hello, I am using a very old textbook from 1895, Loney's Trigonometry, which poses the following problem:

If the circumference of a circle be divided into 5 parts, which are in A.P., and if the greatest part be 6 times the least, find in radians the magnitudes of the angles that the parts subtend at the centre of the circle.

The answers are given as \(\displaystyle \frac{4\pi}{35},\frac{9\pi}{35},\frac{14\pi}{35}, \frac{19\pi}{35},\frac{24\pi}{35}\) radians.

The trouble is that I have no idea what is meant by "5 parts, which are in A.P.". There is no diagram and the only reference to points A and P I can find is in a much earlier diagram, where AP is an arc subtending an angle of one radian. Does anyone have any ideas, based on the specifications of the problem? Thank you!
A.P. means Arithmetic Progression. In other words, each piece differs from the previous one by a constant amount.
 

Ragnarok

Member
Jul 10, 2013
41
Ah, thank you so much! I couldn't imagine what it meant, though it's quite clear now.