Unit Circle
About Unit Circle
A unit circle is defined as a circle with a radius of one unit. A circle is a closed geometric form that has no angles or sides. The unit circle has all of the features of a circle, and its equation is derived from a circle's equation. A unit circle can also be used to calculate the standard angle values of all trigonometric ratios.With the use of trigonometric ratios of cos and sin, we will learn the equation of the unit circle and how to represent each point on the circumference of the unit circle. Get the List of all Maths formulas in one place.
What is Unit Circle?
A unit circle is defined as a circle with a radius of one unit. The cartesian coordinate plane is commonly used to depict the unit circle. The second-degree equation with two variables x and y is used to represent the unit circle algebraically. The unit circle is used in trigonometry to obtain the values of trigonometric ratios such as sine, cosine, and tangent.
Unit Circle Definition
A unit circle is defined as the locus of a point that is one unit away from a fixed point.
Equation of a Unit Circle
A circle's general equation is (x - a)2 + (y - b)2 = r2, which depicts a circle with the centre (a, b) and radius r. This circle equation is simplified to illustrate a unit circle equation. With its centre at (0, 0), the origin of the coordinate axes, and a radius of 1 unit, a unit circle is constructed. As a result, the unit circle's equation is (x - 0)2 + (y - 0)2 =1 2. The equation of a unit circle is obtained by simplifying this.
Equation of the Unit Circle: x2 + y2 = 1
The centre of the unit circle is (0,0), and the radius is 1 unit. All of the locations on the circle in the four quadrants satisfy the above equation.
Finding Trigonometric Functions Using a Unit Circle
A unit circle can be used to calculate the trigonometric functions of sine, cosine, and tangent. To comprehend trigonometric functions, use the Pythagoras theorem in a unit circle. Consider a right triangle in the cartesian coordinate plane, which is centred on a unit circle. The hypotenuse of a right triangle is represented by the circle's radius. The radius vector forms an angle with the positive x-axis, and the radius vector's endpoint coordinates are (x, y). The lengths of the base and altitude of the right triangle are represented by the values of x and y. With the sides 1, x, and y, we now have a right angle triangle. We may obtain the values of the trigonometric ratio by using this in trigonometry.
- sinθ = y/1 = Altitude / Hypoteuse
- cosθ = x/1 = Base / Hypotenuse
We now have sin = y and cos = x, and we can use this to get tan = y/x. Similarly, the right-angled triangle within the unit circle can be used to find the values of the other trigonometric ratios. We can also get the principal values of these trigonometric ratios by modifying the values.
Unit Circle with Sin Cos and Tan
The coordinates(x, y) of any point on the unit circle are equivalent to the trigonometric identities (cos, sin). The cosine and sine of any values made by the radius line with the positive x-axis are represented by the coordinates of the endpoint of the radius. We have cos = x and sin = y, which are useful in calculating the other trigonometric ratio values. By extending this, we get tan = sin/cos or tan = y/x.
Another key element to remember is that the sin and cos values are always between 1 and -1, and the radius value is 1, with a negative x-axis value of -1. The full circle symbolises a 360o angle, while the circle's four quadrant lines indicate 90°, 180°, 270°, and 360° (0). Because the cos value is 0 for 90 and 270, the tan values at these angles are indeterminate.
Unit Circle Chart in Radians
A complete angle of 2π radians is represented by the unit circle. And the unit circle is divided into four quadrants by angles of π/2, 3π/2, and 2π. The standard values, which are appropriate to trigonometric ratios, are further within the first quadrant at the angles of 0, π/6,π/4,π/3,π/2. The standard angle values of the cosine and sine ratios are represented by the points on the unit circle for these angles. The values are repeated across the four quadrants, but with a change in sign, as seen in the diagram below. The reference x-axis and y-axis, which are positive on one side and negative on the other, account for the change in sign.
Unit Circle and Trigonometric Identities
Other trigonometric identities such as cotangent, secant, and cosecant can be obtained using the unit circle identities of sine, cosecant, and tangent. The reciprocal of the sine, cosine, and tangent are the unit circle identities cosecant, secant, and cotangent. Furthermore, we may calculate the value of tanθ by dividing sinθ by cosθ, and the value of cotθ by dividing cosθ by sinθ.
The unit circle identities for a right triangle set in a unit circle in the cartesian coordinate plane with hypotenuse, base, and altitude measuring 1, x, and y units respectively can be written as,
- sinθ = y/1
- cosθ = x/1
- tanθ = sinθ/cosθ = y/x
- sec(θ = 1/x)
- csc(θ) = 1/y
- cot(θ) = cosθ/sinθ = x/y
cot(θ) = cosθ/sinθ = x/y
- 1=sin2θ + cos2θ
- sec2θ =1 + tan2θ
- cosec2θ=1 + cot2θ
Unit Circle and Trigonometric Values
A unit circle can be used to compute the various trigonometric identities and their principal angle values. The x-coordinate of the unit circle is cosine, while the y-coordinate is sine. Let us now calculate their respective values for θ = 0° and θ = 90°.
For θ = 0°, 1 is the x-coordinate, while 0 is the y-coordinate. As a result, cos0 equals 1 and sin0 equals 0. Let's take a look at another 90 angle. Here, cos90 equals 0 and sin90 equals 1. Let us now use this unit circle to calculate key trigonometric function values such as 30, 45, and 60. These values can also be expressed in radians. We know that 360 degrees equals 2 radians. We can now transform angular measurements to radian measurements and express them in radians.
Unit Circle Table???????
Angle θ | Radians | Sinθ | Cosθ | Tanθ = Sinθ/Cosθ | Coordinates |
0° | 0 | 0 | 1 | 0 | (1,0) |
30° | π/6 | 1/2 | √3/2 | 1/√3 | (√3/2, 1/2) |
45° | π/4 | 1/√2 | 1/√2 | 1 | (1/√2, 1/√2) |
60° | π/3 | 1/2 | 1/√2 | √3 | (1/2, √3/2) |
90° | π/2 | 1 | 0 | Undefined | (0,1) |
- secθ = 1/cosθ
- cosecθ = 1/sinθ
- cotθ = 1/tanθ
Unit Circle in Complex Plane
A unit circle consists all complex numbers of absolute value as 1. Therefore, it has the equation of |z| = 1. Complex number z = x + iy will lie on unit circle with equation given as x2 + y2 = 1.
In a complex plane, the unit circle can be thought of as a set of unit complex numbers z provided by the form,
z = e^it = cos t + i sin t = cis (t)
Euler's formula is represented by the above relationship.
Download free pdf of Unit Circle Formula Its Use And Solved Examples
Related Links
- A plus B Whole Square
- A Plus B Whole Cube Formula
- A Minus B Whole Square
- A minus B Whole Cube Formula
- X plus 1 whole cube Formula
- 2cosA cosB Formula
- 3D Distance Formula
- 30-60-90 Formula
- 90 Degree Angle Formula
- a3 plus b3 Formula
- a3 - b3 Formula
- a2+b2+c2 Formula
- Addition Formula
- All Algebra Formula in One sheet
- Algebraic Sequence Formula
- All Circle Formulas
- Altitude of a Triangle Formula
- Amplitude Formula
- Angle Bisector Theorem
- Angle Difference Formula
- Angles Formulas
- Annuity Formula
- ANOVA Test
- Antiderivative Formula
- APY Formula
- Arccot Formula
- Arc Length
- Arctan Formula
- Area of a circle
- Area of a sector of a circle formula
- Area under the curve formula
- Arithmetic Sequence Recursive Formula
- Arithmetic Mean Formula
- Arithmetic Sequence Explicit Formula
- Arithmetic Sequence Formula
- ASA Formula
- Associative Property Formula
- Asymptote Formula
- A Union B Formula
- Average Deviation Formula
- Average Formula
- Average Rate of Change Formula
- Average speed formula
- Axis of the Symmetry
- Basic Algebra Formula
- Cosec Cot Formula
- Baye's Formula
- Binary to Decimal
- Binomial Distribution Formula
- Binomial Expansion
- Binomial Theorem Formula
- Calculus formulas
- Calculus Formulas
- Celsius Formula
- Celsius to Fahrenheit
- Celsius to kelvin
- Centimeter Square to Meter Square Formula
- Central Angle of a Circle Formula
- Central Limit Theorem
- Central Tendency Formula
- Centroid Formula
- Centroid of a Trapezoid Formula
- Chain Rule And Composite Functions
- Change of Base Formula
- Chi-square formula
- Circle Graph Formula
- Clock Angle Formula
- Coefficient of Variation
- Cofactor Formula
- Coin Toss Probability Formula
- Combinations Formula
- Common Ratio Formula
- Commutative Property Formula
- Completing the Square Formula
- Complex Number Formula
- Conditional probability Formula
- Cone Height Formula
- Confidence Interval Formula
- Consecutive Integers Formula
- Continuous Compounding Formula
- Convert Millimeter to Inches Formula
- Correlation Coefficient Formula
- Cos Inverse Formula
- Cos Square Theta Formula
- Cosecant Formula
- Cost Price Formula
- Cot-Tan Formula
- Cotangent formula
- Covariance Formula
- Cpk Formula
- Cross product formula
- Cube formula
- Cube Root Formula
- Cubic Equation Formula
- Degrees to Radians
- Double Angle Formulas
- Quadratic Function
- Quadrilateral Area Formulas
- Quadrilateral Formulas
- Quarterly Compound Interest Formula
- Quartile Formula
- Quarts to liters formula
- Quotient Rule Formula
- R Squared Formula
- Radians to Degrees
- Radical
- Radius of Curvature Formula
- Radius
- Range Formula
- Rate formula
- Rate of Change Formula
- Rate of Return Formula
- Ratio Analysis Formulas
- Ratio Formula
- Rational Numbers Formula
- Reciprocal Formula
- Recursive Formula
- Reduction Formula
- Regular Hexagon Formula
- Relative Change Formula
- Relative Error Formula
- Relative Frequency Formula
- Relative Standard Deviation Formula
- Remainder Formula
- Lagrange Interpolation Formula
- Lateral Area Formula
- Lateral Surface Area of a Cube
- Law of Cosines
- Law of Sines
- Least Common Multiple
- Limit Formula
- Linear Approximation Formula
- Linear Equation Formula
- Linear Function Formula
- Linear Interpolation Formula
- Litres to gallons formula
- Log Formulas
- Factorial
- Factoring Trinomials Formula
- Factorization Formula
- Fahrenheit Equation
- Fahrenheit to Celsius
- Fahrenheit to Kelvin
- Fibonacci Sequence Formula
- Fibonacci Sequence
- Foil Formula
- Foot To Meter Formula
- Formula For 180 Degree Rotation
- Formula For Absolute Value
- Formula For Adding Fractions
- Formula For Adding Consecutive Numbers
- Formula Arc Length
- Formula For Dividing Fractions
- Formula For Finding Angles
- Formula For Odd Numbers
- Formula For Period
- Formula For Squaring
- Isosceles Triangles Formulas
- Formula To Find x Intercept
- Fourier Series Formula
- Fractions Formula
- Frequency Distribution Formula
- Frequency Formula
- Frustum Of A Regular Pyramid Formula
- Frustum of Cone
- Function Formulas
- Function Notation Formula
- Future Value Compound Interest Formula
- Future Value Simple Interest Formula
- G To Kg Formula
- GCD Formula
- GCF Formula
- Geometric Area Formula
- Geometric Distribution Formula
- Geometric Mean Formula
- Geometric Sequence Formulas
- Geometric Series Formula
- Geometric Sum Formula
- Geometry Formulas
- Gradient Of A Line
- Grams To Kilogram Formula
- Great Circle Formula
- Gross Profit Formula
- Half Life Formula
- Harmonic Mean Formula
- HCF Formula
- Height Of A Parallelogram Formula
- Heron's Formula
- Hexadecimal To Decimal Formula
- Hexagon Formula
- How To Convert 20 Celsius Into Fahrenheit
- How To Convert Celsius Into Fahrenheit
- Hyperbola Formula
- Hyperbolic Functions Formula
- Hypotenuse Formula
- Implicit Differentiation Formula
- Independent Events Formula
- Infinite Geometric Series Formula
- Infinite Series Formula
- Integers Formulas
- Integration By Parts Formula
- Integration Formulas
- Integration of UV Formula
- Interest Formula
- Interest Rate Formula
- Interior Angle Formula
- Internal Rate of Return Formula
- Linear Interpolation Formula
- Interquartile Range(IQR) Formula
- Inverse Functions
- Inverse of Matrix
- Inverse Tangent Formula
- Inverse Trigonometric Formulas
- Inverse Variation Formula
- Inversely Proportional Formula
- Perimeter Of The Isosceles Triangle
- Sample Mean Formula
- Sample Size Formula
- Sample Standard Deviation Formula
- Sample Variance
- SAS Triangle Formula
- Scalene Triangle
- Scientific Notation Formula
- Secant Formula
- Selling Price Formula
- Long Division Formula
- Sequences and Series Formulas
- Semicircle Formulas
- Sequence Formulas
- Maclaurin Series Formula
- Magnitude of a Vector
- Sets Formulas
- Magnitude of Vector formula
- P(A/B) Formula
- Matrix Formula
- Mean Absolute Deviation Formula
- Parabola Formula
- Mean Deviation
- Parallel Lines Formula
- Parallelogram Formula
- Mean formula
- Pearson Correlation Formula
- Percent Composition Formula
- Mean Median Mode Formula
- Mean Value Theorem
- Median
- Midpoint Formula
- Miles to Km formula
- Percent Decrease Formula
- Measurement Formulas
- Miles to meter formula
- Percent Error
- Percent Difference Formula
- Percent Yield
- ml to l Formula
- Percentage Increase Formula
- Mode
- Monthly Compound Interest Formula
- Mph to m/s formula
- Percentile Formulas
- Multiple Angle Formulas
- Multiplication
- n Choose k Formula
- Perfect Cube Formula
- Natural Log Formula
- Perfect Square Formula
- NCR Formula
- Perfect Square Trinomial Formula
- Net Change Formula
- Net Income Formula
- Net Present Value Formula
- Perimeter Formulas
- Net Worth Formula
- Newton's Method Formula
- Nominal Interest Rate Formula
- Normal Distribution Formula
- nPr Formula
- Ordinary Annuity Formula
- Orthocenter Formula
- Perimeter Of A Circle Formula
- Z Score
- Perimeter Of A Kite Formula
- Perimeter of a Trapezoid Formula
- Yards to Meters Formula
- Permutation And Combination
- Y-Intercept
- Permutation Formula
- Work Formula
- What are Trigonometric Functions Formulas
- Weight formula
- Perpendicular Line Formula
- Phase Shift Formula
- Weighted Mean
- Weighted Average
- Pi Formula
- Point of Intersection Formula
- Washer Method Formula
- Point Slope Form
- Volume Formulas
- Vieta's Formula
- Venn Diagram Formula
- Vector Projection
- Vector formulas
- Poisson Distribution Formula
- Polygon Formula
- Polynomial Formula
- Variance Formula
- U Substitution Formula
- Unit Vector
- Population Change Formula
- Unit Rate Formula
- Unit Conversion
- Unit Circle
- Uniform Distribution Formula
- Population Mean Formula
- Trigonometry Formulas
- Triangular Pyramid Formula
- Triangle Formulas
- Trapezoidal Rule
- Trajectory Formula
- Total Interest Formula
- Time Formula
- The direction of a vector
- The Accuracy Formula and How to Calculate It
- Temperature Conversion
- T Distribution Formula
- Taylor Series Formula
- Taylor Polynomial Formula
- Tangent Line
- Tangent Formulas
- Tangent Circle Formula
- Tan Formula
- Tan2x Formula
- Surface-area-of-triangular-pyramid-formula
- Surface Area of a Square Prism
- Surface Area Formulas
- Supplementary Angles Formula
- Sum of Squares
- Sum of Perfect Squares Formula
- Sum of n Natural Numbers Formula
- Sum of Integers Formula
- Sum of Exterior Angles Formula
- Sum of Even Numbers Formula
- Sum of Cubes Formula
- Sum of Arithmetic Sequence Formula
- Sum of Angles Formula
- Summation Formulas
- Subtraction
- Stirling Formula
- Standard Form Formula
- Standard Deviation
- SSS Formula
- Square Root
- Square Footage Formula
- Speed formula
- Special Right Triangles
- Special Factoring Formulas
- Slope of the Secant Line Formula
- Slope Intercept Form
- Slope Formula
- Slant Height Formula
- Skewness Formula
- Sin Squared x Formula
- Sin Formula
- Sine Formula
- Sin Cos Tan
- Sin Cos Formulas
- Sin2x Formula
- Simpson's Rule
- Simple Interest
- Similar Triangles Formula
- Signal to Noise Ratio
- Sides of Triangle
- Shell Method Formula
- Rotation Formula
- Root Mean Square Formula
- Rise Over Run Formula
- Right Angled Triangle
- Riemann Sum Formula
- Rhombus Formula
- Reverse Percentages
- Revenue Formula
- Resultant Vector Formula
- Remainder Theorem
- Profit Margin Formula
- Profit Formula
- Profit and loss formula
- Product To Sum Formulas
- Area of Square Formula
- Product Rule
- Area of Trapezoid
- Basic Statistics Formula
- Current Formula
- Daily Compound Interest Formula
- De Moivre's Formula
- Exponential Decay Formula
- Decimal to Binary Formula
- Decimals Formula
- Definite Integral Formula
- Derivative Formula
- Diagonal of a Cube Formula
- Diagonal of a Parallelogram
- Diagonal of a Polygon Formula
- Diagonal of The Rectangle
- Probability Formula
- Prime Numbers Formula
- Diagonal of the Square
- Diagonals Formula
- Prime Factorization Formula
- Present Value Formula
- Present Discounted Value Formula
- Population Variance
- Diameter Formula
- Km to miles formula
- Diameter Of A Sphere Formula Using Volume
- Kilogram to newton formula
- Exterior Angle Theorem
- Difference of Cubes Formula
- Exponential Formula
- Difference of Squares Formula
- Exponential Growth Formula
- Exponential growth and decay formula
- Exponential Function formula
- Exponential Function
- Side Angle Side Formula
- Difference Quotient Formula
- Differential Equation Formula
- Dimensional Formula
- Discriminant Formulas
- Disk Method Formula
- Distance Formula
- Distributive Property Formula
- Exponential Equations
- Dividend Divisor Quotient Remainder Formula
- Exponential Distribution
- Exponential Decay Formula
- Division Formula
- Euler's Formula
- Expected Value Formula
- Dot product
- Euclidean Distance Formula
- Double Angle Formulas
- Equilateral Triangle Formula
- Equation of a circle formula
- Endpoint Formula
- Double Time Formula
- Empirical Rule Formula
- Empirical Probability Formula
- Ellipse Formulas
- Effective Interest Rate Formula
- Effective Annual Rate Formula
- Differentiation and Integration Formula
- Direct Proportion Formula
- Direct Variation Formula
- a2-b2 Formula
- Discount Formula
- Determinant Formula
- Pythagorean Triples Formula
- P-value Formula
- Proportion Formula
- P(A intersections B) Formula
- P(A union B)Formula
- Decimal to Hexadecimal
- Vertex Formula
- Area of Octagon
- Outlier Formula
- Distance Between Two Points Formula