"In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1."
A circle of radius 1 centered at the origin of a coordinate system, used to visualize the values of trigonometric functions for angles in all quadrants.
Angles and Degrees: A basic understanding of angles and how they are measured in degrees is essential for trigonometry.
Radians: In trigonometry, radians are used to measure angles in a more precise way than degrees.
Unit Circle: The Unit Circle is a circle with a radius of 1 that is used to visualize trigonometric functions.
Y-axis and X-axis: The Y-axis and X-axis are the vertical and horizontal lines that intersect at the origin of a coordinate plane.
Sine, Cosine, and Tangent: These are the three primary trigonometric functions, used to calculate the side lengths of right triangles.
Cotangent, Secant, and Cosecant: These are the other three trigonometric functions, derived from the primary functions.
Reference Angles: Reference angles are angles that are less than 90 degrees and are used to simplify calculations in trigonometry.
Trigonometric Functions of Special Angles: Knowing the values of sine, cosine, and tangent for 30, 45, and 60 degrees is crucial in trigonometry.
Finding the Side Lengths of Triangles: Trigonometry can be used to calculate the lengths of the sides of right triangles, given certain information.
Graphing Trigonometric Functions: Understanding how to graph sine, cosine, and tangent functions is an important skill in trigonometry.
The Law of Sines: The Law of Sines is used to solve non-right triangles, where all three angles are known.
The Law of Cosines: The Law of Cosines is used to solve non-right triangles, where at least one side length is unknown.
Inverse Trigonometric Functions: Inverse trigonometric functions are used to calculate the angle given the ratios of sides.
Trigonometric Identities: Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved.
Applications of Trigonometry: Trigonometry is used in many fields, including physics, engineering, and astronomy, as well as in everyday life.
Unit Circle with Degrees: This is a type of Unit Circle where angles are measured in degrees instead of radians. The circle is divided into 360 degrees, with each degree representing a specific point on the circle.
Unit Circle with Radians: This is a conventional Unit Circle where angles are measured in radians. A full circle is divided into 2π radians, and each radian represents a specific point on the circle.
Unit Circle with Cosine: This type of Unit Circle represents the cosine values of angles on the circle. Each point on the circle corresponds to a specific cosine value, and the values are plotted against the x-axis.
Unit Circle with Sine: Like the Unit Circle with Cosine, this type of Unit Circle represents the sine values of angles on the circle. Each point on the circle corresponds to a specific sine value, and the values are plotted against the y-axis.
Unit Circle with Secant: This is a type of Unit Circle that represents the secant values of angles on the circle. The secant is calculated by dividing the hypotenuse by the adjacent side of a right triangle, and each point on the circle corresponds to a specific secant value.
Unit Circle with Cosecant: Similar to the Unit Circle with Secant, this type of Unit Circle represents the cosecant values of angles on the circle. The cosecant is the reciprocal of the sine, so each point on the circle corresponds to a specific cosecant value.
Unit Circle with Tangent: This type of Unit Circle represents the tangent values of angles on the circle. The tangent is calculated by dividing the opposite side of a right triangle by the adjacent side, and each point on the circle corresponds to a specific tangent value.
Unit Circle with Cotangent: This is a type of Unit Circle that represents the cotangent values of angles on the circle. The cotangent is the reciprocal of the tangent, so each point on the circle corresponds to a specific cotangent value.
"The unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system."
"In topology, it is often denoted as S1 because it is a one-dimensional unit n-sphere."
"If (x, y) is a point on the unit circle's circumference, then |x| and |y| are the lengths of the legs of a right triangle whose hypotenuse has length 1."
"Thus, by the Pythagorean theorem, x and y satisfy the equation..."
"...the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant."
"The interior of the unit circle is called the open unit disk."
"The interior of the unit circle combined with the unit circle itself is called the closed unit disk."
"One may also use other notions of 'distance' to define other 'unit circles', such as the Riemannian circle; see the article on mathematical norms for additional examples."
"A unit circle has a radius of 1."
"The unit circle is centered at the origin (0, 0) in the Cartesian coordinate system."
"The equation represents a right triangle with x and y as the lengths of the legs and the hypotenuse having a length of 1."
"The above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant."
"The interior of the unit circle is known as the open unit disk."
"The closed unit disk includes both the interior of the unit circle and the unit circle itself."
"Other 'unit circles' can be defined using different notions of 'distance,' such as the Riemannian circle."
"The radius of a unit circle is set to 1."
"The origin (0, 0) serves as the center of the unit circle."
"The equation x^2 = (-x)^2 holds for all x on the unit circle."
"The equation x^2 + y^2 = 1 is a representation of the unit circle, but other circles can be defined using different mathematical norms."