Find angles when circumference is divided into 5 unequal parts

• MHB
• Ragnarok7
In summary, the problem is asking for the angles subtended at the center of a circle when its circumference is divided into 5 parts in an Arithmetic Progression, with the greatest part being 6 times the smallest. The answers are given in radians as \frac{4\pi}{35},\frac{9\pi}{35},\frac{14\pi}{35}, \frac{19\pi}{35},\frac{24\pi}{35}. The only confusion was with the term "A.P." which stands for Arithmetic Progression.
Ragnarok7
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!

Ragnarok said:
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.

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

1. How do I find the measure of each angle when the circumference is divided into 5 unequal parts?

To find the measure of each angle, you will need to first determine the total measure of the circumference, which is 360 degrees. Then, divide 360 by 5 to get the measure of each equal part, which is 72 degrees. However, since the parts are unequal, you will need to use proportions or ratios to find the measure of each angle.

2. Can I use a protractor to measure the angles?

Yes, you can use a protractor to measure the angles. However, since the parts are unequal, you will need to use a ruler to measure the lengths of the parts and then use the protractor to measure the corresponding angles.

3. What is the formula for finding the measure of each angle when the circumference is divided into 5 unequal parts?

The formula for finding the measure of each angle is (360/Total circumference) x Length of part. This formula uses proportions to find the measure of each angle based on the length of the part in comparison to the total circumference.

4. Can I use the same formula for finding the measure of angles if the circumference is divided into a different number of unequal parts?

Yes, you can use the same formula for finding the measure of angles if the circumference is divided into a different number of unequal parts. You will just need to adjust the formula to correspond with the number of parts, such as dividing 360 by the number of parts instead of 5.

5. What if I don't know the length of the parts in the circumference?

If you do not know the length of the parts in the circumference, you will need to use proportions to compare the lengths of the parts in relation to the total circumference. Then, you can use the formula (360/Total circumference) x Length of part to find the measure of each angle.

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