Unit Circle Formula

Unit Circle Formula

Extramarks experts have provided an explanation of the Unit Circle Formula along with a solved example problem. To refresh students’ memories, a Unit Circle Formula in mathematics is a circle having a radius of one. The Unit Circle Formula, particularly in trigonometry, is the circle of radius one that is centred at (0, 0) on the Euclidean plane in the Cartesian coordinate system.

The generalisation to higher dimensions is the unit sphere, while the unit circle is sometimes abbreviated S1. The Unit Circle Formula interior is referred to as the “closed unit disc,” whereas the interior of the unit circle and the unit circle itself are referred to as the “open unit disc.”

A circle with a unit radius is what is meant by the word “Unit Circle Formula.” The closed geometric shape of a circle lacks all edges and angles. The Unit Circle Formula possesses all of a circle’s characteristics, and its equation is also derived from a circle’s equation. A Unit Circle Formula may also be used to calculate the standard angles for all trigonometric ratios.

Here, using trigonometric cosine and sine ratios, students will learn the equation for the Unit Circle Formula and comprehend how to represent each point on its circumference.

What is Unit Circle?

A circle with a radius of one unit is referred to as a Unit Circle Formula. In the cartesian coordinate plane, the Unit Circle Formula is often shown. The second-degree equation with the variables x and y is used to algebraically represent the Unit Circle Formula. The trigonometric ratios sine, cosine, and tangent may all be calculated using the Unit Circle Formula, which has applications in trigonometry.

Students can access these notes and solutions centred on the Unit Circle Formula made available on the Extramarks website and mobile application for students and teachers. The Unit Circle Formula notes and solutions are extremely student friendly and easy to comprehend. The framework of the Unit Circle Formula notes and solutions has a very easy to understand framework, making sure that students understand the concept and retain it. The Unit Circle Formula has been compiled in collaboration with the top experts at Extramarks, thereby eliminating any chance of error or inaccuracy in the solutions.

Students will therefore not need to worry about the authenticity of the notes provided by Extramarks experts.

Unit Circle Definition

A unit circle is the location of a point that is one unit away from a fixed point.

Experts at Extramarks have kept the definition of the Unit Circle Formula short and simple, so students need not worry about learning lengthy definitions and forgetting about them later.

Equation of a Unit Circle

(x – a)2 + (y – b)2 = r2 is the general equation for a circle, which depicts a circle with centres (a, b) and a radius (r). To show the equation of a Unit Circle Formula, this circle’s equation has been reduced. With its centre at the coordinate axes’ starting point (0, 0) and a radius of 1 unit, a unit circle is created. As a result, the unit circle’s equation is (x – 0)2 + (y – 0)2 = 12. To get the equation of a unit circle, this is simplified.

Unit Circle Equation: x2 + y2 = 1.

The radius of the unit circle in this instance is one unit, and its centre is (0, 0). All of the circle’s points are satisfied by the aforementioned equation.

Finding Trigonometric Functions Using a Unit Circle

What connects the trigonometric functions sine, cosine, and tangent to the unit circle is called the Unit Circle Formula. Actually, a circle with radius 1 hanging in a particular quadrant of the coordinate system is referred to as the Unit Circle Formula. A unit circle’s radius may be measured at any location along its edge.

A right-angled triangle is created. This Unit Circle Formula angle will be shown via the angle indicator. The two control points would only need to be clicked and moved if a student wanted to modify their grade.

Unit Circle with Sin Cos and Tan

• Sine

Sine is a fundamental trigonometric function that is represented by the symbol.Sine is calculated mathematically by dividing the hypotenuse of a right-angled triangle by its perpendicular. With the aid of the aforementioned connection, we may calculate the angle or length of any construction. Consequently, the following is the formula to compute Sine:

Perpendicular/hypotenuse = Sine

Cosecant: The reciprocal of sine with regard to cosine is known as cosecant. It may be calculated either by dividing the sine by one or by reciprocating the sine. cosecant, thus equals 1/sin.

• Cosine

In a right-angled triangle, the ratio between the base and hypotenuse of a triangle is referred to as cosineθ. It is actually one of the most crucial trigonometric functions of all. In Mathematical terms, cosine is obtained by dividing the base of a right-angled triangle with its hypotenuse. Hence, the formula to calculate Cosineθ is as below;

Cosine = Base/Hyp

Secant: The reciprocal of cosine which is known as secant θ is also used in some triangles. The secant θ is used in several numerical calculations and is calculated by reciprocating the cosine θ. Thus, Secant = 1/cosine.

• Deviation

Tangent is the third and final basic trigonometric function. In a right-angled triangle, the tangent may be calculated in the same way as sine and cosine are. In a right triangle, we can simply get the value of the tangent by dividing the triangle’s perpendicular by its base.

Tangent is calculated mathematically using Tang = Perp/Base.

Cot: The inverse of Tangent is called cot. The value of coe can be calculated by multiplying the value of tangent by itself.This equation’s mathematical form is as follows: Cot equals 1/Tang.

The interactive unit circle graph may be used to understand all equations and trigonometric functions.

Unit Circle Chart in Radians

To start, the radian is a unit for measuring angles that we mostly use in trigonometry. We substitute it with degrees. A whole sphere or circle is slightly over six radians, as opposed to 360 degrees for a full sphere or circle.

An SI unit that is useful in measuring angles is the radian. Additionally, it is the accepted unit of measurement for angular distance in all areas of mathematics. The measurement in radians of the angle that an arc subtends is numerically equal to the length of the arc in the unit circle.

Just below 57.3 degrees is one radian, which extends to OEIS: A072097. The unit, however, was previously a SI unit, according to the addendum. The radian has been an SI-derived unit since 1995.

Separately, the solid angle measurement uses the steradian as the SI unit. Furthermore, the sign rad is most commonly used to represent the radian.C serves as its alternate symbol. the letter r, a superscript, or the letter c in superscript.

Unit Circle and Trigonometric Identities

The circle with radius 1 and a centre at the origin is known as the Unit Circle Formula. The unit circle’s equation is x2 + y2 = 1. In the lesson, t denotes an angle that is measured in the opposite direction of the positive x-axis. As demonstrated in the picture on the Extramarks website and mobile application, students shall define (x, y) as the location where the ray at angle t crosses the unit circle for a given value of t.

Unit Circle Pythagorean Identities

An equation that holds true for all potential values is called an identity in mathematics. “Trigonometric identity” is the name given to an equation involving trigonometric functions that is true regardless of the value used to replace the variable. For that value, we suppose that both sides are “defined.” Particularly helpful for making trigonometric formulas simpler are trigonometric identities. The Pythagorean Theorem-related trigonometric identities are the most significant. Students will discover what Pythagorean Identities are in the article that is available on the Extramarks website and mobile application.

Students can learn the connection between the Unit Circle Formula and the Pythagorean Identities on the Extramarks website and mobile application. The solutions and notes for the Unit Circle Formula are downloadable and can be used for offline study. The Unit Circle Formula can help students strengthen their basics of the subject and facilitate self-studying. Students can also clarify their doubts using the Unit Circle Formula notes and solutions offered by Extramarks experts.

Unit Circle and Trigonometric Values

The Unit Circle Formula may be used to determine the trigonometric function for the primary values. If the radius of a Unit Circle Formula with a centre at (0, 0) and a radius of 1 unit is inclined at an angle of and the endpoint of the radius vector is (x, y), then cos(x, y) and sin(x, y) are equal. These two numbers may be used to determine all other trigonometric ratios. Additionally, by modifying the value of, the primary values may be calculated.

Unit Circle Table:

The Unit Circle Table is available for student access on the Extramarks website and mobile application. Students can learn more about the unit circle table, review it, and clarify their doubts before their examinations.

The Unit Circle Formula has been mentioned and highlighted by the Extramarks experts. These notes and solutions based on the Unit Circle Formula are extremely helpful from an exam perspective. Examples citing and explaining the Unit Circle Formula have been provided wherever needed throughout the Unit Circle Formula solutions by Extramarks experts. Experts at Extramarks have also made sure to provide students with high quality illustrations and diagrams throughout the article containing the solutions.

Unit Circle in Complex Plane

The Unit Circle Formula is also available in Hindi. Mathematics can be a little too tough for students, but they need not worry as long as they religiously follow the notes and solutions centred on the Unit Circle Formula provided by Extramarks experts. These solutions are surely going to help students improve their performance in examinations. The solutions are comprehensive, detailed, and descriptive, thereby ensuring that students fully understand what they are meant to learn through the solutions.

Unit Circle Examples

Example 1: Does the point P (1/2, 1/2) lie on the unit circle?

Solution:

It is known that the equation of a unit circle is:

x2 + y2 = 1

Substituting x = 1/2 and y = 1/2, we get:

= x2 + y2

= (1/2)2 + (1/2)2

= 1/4 + 1/4

= 1/2

≠ 1

Since, x2 + y2 ≠ 1, the point P (1/2, 1/2) does not lie on the unit circle.

Answer: Therefore (1/2, 1/2) doesn’t lie on the unit circle.

Practice Questions on Unit Circle

Students following Extramarks can find a series of practice questions based on the Unit Circle Formula on the Extramarks website and mobile application. Practising these questions will help them be more inclined towards self-study. Extramarks experts have also made sure to provide curated assessments to evaluate the progress of students. These solutions cater to the needs of students at the individual level as well as at the national and local level.