Visualizing a Segment of a Unit Circle

In summary, the given commands will draw a segment of a unit circle from 60 degrees to 120 degrees. The equation rho1 = rho*sin(a1)./sin(theta) is used to convert from polar to Cartesian coordinates and create a straight line. The equation represents a rational function with vertical asymptotes at x=2 and x=3 and a horizontal asymptote at y=2.
  • #1
chrisych
32
0
a = 60*pi/180;
a1 = (pi - a)/2;
a2 = (pi + a)/2;
theta = a1: a/60: a2;
rho = ones(size(theta));
rho1 = rho*sin(a1)./sin(theta);
polar(theta, rho);
hold on;
polar(theta, rho1)

03.jpg


The above commands will draw a segment of a unit circle which starts from 60[tex]^{o}[/tex] to 120[tex]^{o}[/tex].

I know everything except the line of "rho1 = rho*sin(a1)./sin(theta);" that I don't know why this equation works from 1 --> 0.866 --> 1 and becomes a straight line. Can anyone tell me the reason? Is it related to a similar triangle or inverse proportion?
 
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  • #2
try converting from polar to euclidean/cartesian...to see if you can figure out how they got that straight line. Remember what does rho,rho1 stand for in polar.
 
  • #3
Cartesian to Polar:
r = sqrt(x^2 + y^2)
theta = arctan(y/x)

Polar to Cartesian:
x = r cos theta
y = r sin theta

rho = 1 (for unit circle)

rho1 = rho*sin(a1)/sin(theta) = 1*sin(60)/sin(60) to 1*sin(60)/sin(120) = 1 --> 0.866 --> 1
 
Last edited:
  • #4
Write the equation of a rational function that has vertical asymptotes at x=2 and x=3 and a horizontal asymptote at y=2

yeah, i really need help with this problem. Thanks so much
 

What is a unit circle?

A unit circle is a circle with a radius of 1 unit that is centered at the origin of a coordinate system. It is commonly used in mathematics and physics to understand the relationships between angles and trigonometric functions.

What is the significance of visualizing a segment of a unit circle?

Visualizing a segment of a unit circle can help us understand the properties of triangles and how they relate to trigonometric functions. It can also help us solve problems involving angles and their corresponding values on the unit circle.

How do you visualize a segment of a unit circle?

To visualize a segment of a unit circle, you can use a protractor to measure the angle and then plot the corresponding point on the circle. Alternatively, you can use a graphing calculator or software to generate a graph of the unit circle.

What is the relationship between angles and trigonometric functions on a unit circle?

The relationship between angles and trigonometric functions on a unit circle is that the coordinates of a point on the unit circle correspond to the values of the sine, cosine, and tangent functions of the angle formed by the radius of the circle and the x-axis.

Why is visualizing a segment of a unit circle important in real-world applications?

In real-world applications, visualizing a segment of a unit circle is important because it allows us to understand and solve problems involving angles and trigonometric functions. This can be useful in fields such as engineering, physics, and navigation.

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