What is a unit circle in trigonometry, and how is it used to define the trigonometric functions?

Answer 1

A unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The equation of the unit circle is given by:

$ x^2 + y^2 = 1 $

In trigonometry, the unit circle is used to define the trigonometric functions sine and cosine. For a given angle $ \theta $, measured from the positive x-axis, the coordinates of the corresponding point on the unit circle are $(\cos(\theta), \sin(\theta))$. These definitions can be extended to all real numbers by considering the angle $ \theta $ to be the result of wrapping the real line around the unit circle.

Furthermore, the unit circle allows us to define the other trigonometric functions as follows:

• Tangent: $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $
• Cosecant: $ \csc(\theta) = \frac{1}{\sin(\theta)} $
• Secant: $ \sec(\theta) = \frac{1}{\cos(\theta)} $
• Cotangent: $ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} $

Answer 2

The unit circle is a circle with a radius of 1 unit centered at the origin (0,0) in the coordinate plane. Its equation is:

$ x^2 + y^2 = 1 $

It serves as an essential tool in trigonometry. For an angle $ heta $ drawn from the positive x-axis, the coordinates of the intersection of the terminal side of the angle with the unit circle are $(cos( heta), sin( heta))$. This representation helps define the trigonometric functions for all angles by wrapping the angle around the circle.

The unit circle also allows the definition of other trigonometric functions:

• $ an( heta) = frac{sin( heta)}{cos( heta)} $
• $ csc( heta) = frac{1}{sin( heta)} $
• $ sec( heta) = frac{1}{cos( heta)} $
• $ cot( heta) = frac{cos( heta)}{sin( heta)} $

Answer 3

A unit circle is a circle with a radius of 1, centered at the origin. Its equation is:

$ x^2 + y^2 = 1 $

The unit circle defines sine and cosine. For an angle $ heta $, the coordinates are $(cos( heta), sin( heta))$. Other functions are:

• $ an( heta) = frac{sin( heta)}{cos( heta)} $
• $ csc( heta) = frac{1}{sin( heta)} $
• $ sec( heta) = frac{1}{cos( heta)} $
• $ cot( heta) = frac{cos( heta)}{sin( heta)} $

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