# Why does the Unit Circle work?

• physicsdreams
In summary, the unit circle and its relationship to the various trigonometric functions were defined based on triangles. The radius of the circle must be 1 for the circle to work, as the hypotenuse of the right triangle formed by a point on the circle and the center of the circle must be 1 to make the opposite side equal to the sine of the angle. The concept of trigonometry was first developed in India and then expanded upon by Islamic Arabian mathematicians. It was later seen that the trigonometric functions could also be defined using the unit circle, and this connection can be seen through the use of right triangles.
physicsdreams
In my Trig class, we learned about the unit circle and its relationship to the various trig functions (sin, cos, etc.).

I am curious to know why the unit circle works the way it does, and the how it was "derived" so to speak.

More specifically, why does radius of the circle have to be 1 for the circle to work.

Please try to keep your explanations as mathematically friendly as possible, as I am only in precalculus.

Thank you

All of those functions we're defined based on the unit circle. It works because we designed it to work.

Number Nine said:
All of those functions we're defined based on the unit circle. It works because we designed it to work.

I think the sine was defined as the opposite side divided by the hypotenuse, in a right angled triangle. I'm pretty sure sin and cos were used before cartetsian coordinates.

Since the hypotenuse will be the radius of the unit circle, this has to be 1 to make the length of the side opposite to angle A equal to sin(A)

willem2 said:
I think the sine was defined as the opposite side divided by the hypotenuse, in a right angled triangle. I'm pretty sure sin and cos were used before cartetsian coordinates.

Since the hypotenuse will be the radius of the unit circle, this has to be 1 to make the length of the side opposite to angle A equal to sin(A)

very sensible,,,
Thanks for the explanation,,,

The idea begins When Babylonians have developed a system for measuring angles "degree measure", They discovered some trigonometric theorems but not explicitly , and then the sin function was first developed in India . But trigonometry wasn't translated to Europe through the Hindus , but through Muslims " arab " . Islamic Arabian mathematicians had taken the ideas from the hindus and expanded them where they have known the six main trigonometric function , they were Calculating them using triangles . So the trigonometry was invented depending on triangles , but it was seen after that its relation with unit circle which willem2 had mentioned . All I can say you that with that curiosity and when you take integral Calculus , I advise you to read spivak's construction for trigonometric function which is even more rigorous than defining them in terms of unit circle , where you will see why .

if you are asking about the connection between the "circular" functions (sine and cosine as defined using the unit circle) and the "trigonometric functions) (sine and cosine as defined using right triangles), from a point, (x, y), on the unit circle, draw the line from (x, y) to (0, 0), the line from (x, y) to (x, 0), and the line from (x, 0) to (0, 0).

That gives a right triangle which has "near side" of length x, "opposite side" of length y, and hypotenuse of length 1. Therefore, $sin(\theta)=$"opposite side over hypotenuse= y/1= y and $cos(\theta)=$"near side over hypotenuse"= x.

(Since the angles in a triangle can only be between 0 and 90 degrees, we really have to assume that (x, y) is in the first quadrant. The circle definition extends that to all real numbers.)

## 1. Why is the Unit Circle used in trigonometry?

The Unit Circle is used in trigonometry because it provides a visual representation of the relationship between the angles and the trigonometric functions sine, cosine, and tangent. It simplifies calculations and makes it easier to understand the properties of these functions.

## 2. How does the Unit Circle work?

The Unit Circle is a circle with a radius of 1 unit and its center at the origin of a coordinate plane. The angles on the Unit Circle are measured in radians, and the coordinates of any point on the circle can be found using the trigonometric functions. This allows for the visualization and understanding of the relationships between the angles and the trigonometric functions.

## 3. What is the significance of the coordinates on the Unit Circle?

The coordinates on the Unit Circle represent the values of the sine and cosine functions for a given angle. The x-coordinate is the cosine value, while the y-coordinate is the sine value. This relationship is important in solving trigonometric equations and understanding the properties of these functions.

## 4. How is the Unit Circle related to right triangles?

The Unit Circle is closely related to right triangles through the trigonometric functions. The sine, cosine, and tangent of an angle in a right triangle can be found by using the ratios of the side lengths to the hypotenuse. These same ratios can be represented on the Unit Circle, allowing for the visualization of the relationship between the angles and the trigonometric functions.

## 5. Why is the Unit Circle considered a fundamental concept in trigonometry?

The Unit Circle is considered a fundamental concept in trigonometry because it is used as a basis for understanding the properties and relationships of the trigonometric functions. It simplifies calculations and allows for a visual representation of these concepts, making it an essential tool in solving trigonometric equations and problems.

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