Circle Equation Help: Finding the Area and Angle Using the Cosine Rule

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In summary, the conversation is about finding the small area between a circle and the y-axis and using calculus to calculate the area. The second question involves finding the angle subtended by a chord on the circle using the cosine rule. The conversation also touches on the concept of explicitation and using a function of "y" to calculate the area. There is also discussion about notation and the difference between "loose/sloppy" and "rigorous/careful" notation.
  • #1
roger
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hi, I am stuck on this question for homework :

I'm not sure how to find the answer, as I thought integration is used only if the curve intersects at the x-axis ? :confused:


Thanx for any help


1.) The equation of circle x^-8x+y^2-10y=-16

It intersects the y-axis at two points : (0,2) and (0,8)

Find the small area between the curve and the y axis.(on the left of the y axis)


2.) Also, the chord between 0,2 and 0,8 on the circle subtends an angle at the centre (4,5).

Show that the angle is 7/25 using the cosine rule.


I'm not sure what to do here as well..
 
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  • #2
Plot that circle and then decide whether u'll be needing calculus to find that area.

As for the second,i don't follow.Is it the same circle...?

Daniel.
 
  • #3
dextercioby said:
Plot that circle and then decide whether u'll be needing calculus to find that area.

As for the second,i don't follow.Is it the same circle...?

Daniel.


The area is bound between the y-axis and the circle. But I don't see how else it could be done without calculus ?

The second question is the same circle and I need to find the angle .
(0,2) and (0,8) is the chord on the circle .
 
  • #4
Alright,then,write the equation and see which branch of the explicitation will u be needing (hint:u can explicitate "x" as a function of "y",because you're given the intercepts with the Oy axis).

Daniel.
 
  • #5
dextercioby said:
Alright,then,write the equation and see which branch of the explicitation will u be needing (hint:u can explicitate "x" as a function of "y",because you're given the intercepts with the Oy axis).

Daniel.


I've not heard that word before...explicitate ..

but I don't understand what your saying.
 
  • #6
Hmmm,alright.Your equation's typically

[tex] (x-x_{0})^{2}+(y-y_{0})^{2}=1 [/tex]

Express x=x(y).

Daniel.
 
  • #7
dextercioby said:
Hmmm,alright.Your equation's typically

[tex] (x-x_{0})^{2}+(y-y_{0})^{2}=1 [/tex]

Express x=x(y).

Daniel.


I don't get it :grumpy:


especially
Express x=x(y).
 
  • #8
You need to put somethting,a function,in this integral

[tex] \mbox{Area}=\int_{\mbox{intercept no.1}}^{\mbox{intercept no.2}} f(y) \ dy [/tex]

And that function of "y" (f(y)) is the one u'll be getting from the circle's equation.

Daniel.
 
  • #9
dextercioby said:
You need to put somethting,a function,in this integral

[tex] \mbox{Area}=\int_{\mbox{intercept no.1}}^{\mbox{intercept no.2}} f(y) \ dy [/tex]

And that function of "y" (f(y)) is the one u'll be getting from the circle's equation.

Daniel.


okay, and then take away the the intercept 2 - 1 ?

but what does x(y)=x mean ? I need a thorough explanation on this please.
 
  • #10
It's a notation;x=x(y) means the function "x" written in terms of the variable "y".

Daniel.
 
  • #11
dextercioby said:
It's a notation;x=x(y) means the function "x" written in terms of the variable "y".

Daniel.


but if i took away the 1st from the second integral is the order important ?

what difference does it make ?

If the area was not bound by either x or y-axis is there a direct way to calulate the area?
 
  • #12
Yes,of course.The area of a plane domain [itex] D\subseteq\mathbb{R}^{2} [/itex] is given by

[tex] S_{D}=:\iint_{D} dx \ dy [/tex]

as long as u choose Oxy as a set of orthormal coordinates in the plane.

Daniel.
 
  • #13
dextercioby said:
It's a notation;x=x(y) means the function "x" written in terms of the variable "y".

Daniel.


Daniel, what difference would it make if I wrote g(y)=x ?
 
  • #14
roger said:
Daniel, what difference would it make if I wrote g(y)=x ?
The difference is that x=x(y) is a "loose/sloppy" notation, whereas x=g(y) is an example of a "rigorous/careful" notation. That's basically it.
 

1. What is the cosine rule?

The cosine rule, also known as the law of cosines, is a mathematical formula used to find the side lengths or angles of a triangle when given two sides and the included angle or all three sides of the triangle. It is based on the cosine function in trigonometry.

2. How do I use the cosine rule to find the area of a circle?

To find the area of a circle using the cosine rule, you must first use the formula to find the length of the radius of the circle. Then, you can use the standard formula for the area of a circle, A = πr², where A is the area and r is the radius, to calculate the area.

3. Can the cosine rule be used to find the angle of a circle?

Yes, the cosine rule can be used to find the angle of a circle. It can be used to find the central angle of a circle when given the length of the radius and the length of an arc. It can also be used to find the angle formed by two intersecting chords in a circle.

4. What is the difference between the cosine rule and the Pythagorean theorem?

The cosine rule and the Pythagorean theorem are both mathematical formulas used to find the lengths of sides in a triangle. However, the Pythagorean theorem can only be used for right triangles, while the cosine rule can be used for any type of triangle. The Pythagorean theorem also only involves the lengths of two sides, while the cosine rule involves the lengths of all three sides or two sides and the included angle.

5. Are there any limitations or restrictions when using the cosine rule?

Yes, there are a few limitations and restrictions when using the cosine rule. It can only be used for triangles, and it is most useful when at least one side or angle is known. It also may not be accurate for very small or very large angles, as the cosine function approaches 0 or 1, respectively. Additionally, it may not work for certain types of triangles, such as those with very short sides or very obtuse angles.

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