Trigonometry: finding an angle, area and length of sector of a circle

In summary, the person does not think that the formula they found in a textbook is correct. They are unsure if they have missed a detail or overlooked a necessary component of the question. They are looking for guidance on how to solve the problem.
  • #1
AN630078
242
25
Homework Statement
Hello I have come across the question below while revising some trigonometry problems, however, I think I have misunderstood the question because I feel that I have too easily arrived at solutions. I must have missed something or be omitting a method to my workings, I would be very grateful of any guidance 👍

A circle has centre O and a radius of 8cm. The arc AB on the circumference is 11cm long.

1. What is the size of angle AOB in radians?
2. What is the area of the sector AOB?
3. Find the length of the line AB.
Relevant Equations
s= θr
A=1/2r^2θ
1. Using the formula for the arc length; s= θr
I have endeavoured to find the angle AOB sine both the arc length and radius are known;
11= θ*8
θ=11/8=1.375 rad

I actually do not think that this can be correct as it seem to simplistic a response. Have I misinterpreted the question or used the wrong formula? Would it be that the arc length is not 11cm?
Is there a more appropriate method I could adopt to concisely answer this question?

2. Area of a sector = 1/2r^2 θ
This will only be correct if θ is indeed 1.375 rad then,
A=1/2*8^2*1.375=44 cm^2

Again, is there a better approach I could take here?

3. So to find the length of the line AB is would this be the base forming a triangle AOB. If we divide this into two forming triangle AOD, where the midpoint between A and B is D then one can use trigonometry to find the side AD.
The known angle is 1.375 rad / 2 = 0.6875 rad.
sin 0.6875 = AD/8
AD=8*sin 0.6875
AD=5.07685... cm
Since AB = 2AD
AB=2*5.07685~10.2 cm to 3.s.f

I feel that I have oversimplified all of the above problems and arrived at too easy a solution, surely I must be doing something wrong? OI would be very grateful for any guidance.
 

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  • #2
AN630078 said:
surely I must be doing something wrong?
Why this lack of confidence ? It's an introductory exercise. Check your work and be confident if it appears OK !
 
  • #3
BvU said:
Why this lack of confidence ? It's an introductory exercise. Check your work and be confident if it appears OK !
Thank you for your reply. Well to be perfectly candid the textbook I found it in also accompanied how many marks each section was worth. For example for question 3 that is supposedly worth 5 marks but I cannot see the level of my workings warranting five points. I thought perhaps that I had missed a crucial detail or overlooked a necessary component of the question. Perhaps it is that my method is not applicable here and that there is a more fitting alternative?
 

Related to Trigonometry: finding an angle, area and length of sector of a circle

1. What is trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is used to solve problems involving right triangles, circles, and periodic functions.

2. How do you find an angle using trigonometry?

To find an angle using trigonometry, you can use the sine, cosine, or tangent ratios. These ratios relate the sides of a right triangle to its angles. You can use a calculator or trigonometric tables to find the value of the ratio and then use inverse trigonometric functions to find the angle.

3. How do you find the area of a sector of a circle using trigonometry?

To find the area of a sector of a circle using trigonometry, you can use the formula A = (θ/360)πr², where θ is the central angle of the sector and r is the radius of the circle. You can also use the formula A = (1/2)r²θ, where θ is in radians.

4. How do you find the length of a sector of a circle using trigonometry?

To find the length of a sector of a circle using trigonometry, you can use the formula L = (θ/360)2πr, where θ is the central angle of the sector and r is the radius of the circle. You can also use the formula L = rθ, where θ is in radians.

5. How is trigonometry used in real life?

Trigonometry is used in various fields such as engineering, physics, astronomy, and navigation. It is used to solve problems involving distances, heights, angles, and trajectories. For example, it is used in surveying to measure distances and angles, in architecture to design and construct buildings, and in astronomy to calculate the position and movement of celestial objects.

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