5.2.1
Unit Circle
The Unit Circle
The Unit Circle
A unit circle is a circle centered at the origin with a radius of 1.
![Illustrative background for Radians](https://image-v2.cdn.app.senecalearning.com/2020-01/c052aacf-2add-4104-a432-730fba7ffed6/Screenshot%202020-01-24%20at%2017.45.49-min,h_400,q_80,w_640.png)
![Illustrative background for Radians ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-01/c052aacf-2add-4104-a432-730fba7ffed6/Screenshot%202020-01-24%20at%2017.45.49-min,h_400,q_80,w_640.png)
Radians
Radians
- An angle of one radian is created at the center of a unit circle by two radii and an arclength of 1.
- The arclength of a unit circle is then equal to the angle .
- This means we can write a general point on the circle as a function of .
![Illustrative background for Trigonometric functions](https://image-v2.cdn.app.senecalearning.com/2020-01/4ee60df3-f667-4124-83fa-314102672ef2/Screenshot%202020-01-24%20at%2017.45.46-min,h_400,q_80,w_640.png)
![Illustrative background for Trigonometric functions ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-01/4ee60df3-f667-4124-83fa-314102672ef2/Screenshot%202020-01-24%20at%2017.45.46-min,h_400,q_80,w_640.png)
Trigonometric functions
Trigonometric functions
- The cosine function is defined as the function that takes the angle from the positive -axis as its input and gives the
-coordinate of a point on the unit circle:
- Likewise, the sine function is defined as the function that takes the same angle and gives the corresponding -coordinate of the point on the unit circle.
![Illustrative background for Sine and cosine](https://image-v2.cdn.app.senecalearning.com/2020-01/4ee60df3-f667-4124-83fa-314102672ef2/Screenshot%202020-01-24%20at%2017.45.46-min,h_400,q_80,w_640.png)
![Illustrative background for Sine and cosine ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-01/4ee60df3-f667-4124-83fa-314102672ef2/Screenshot%202020-01-24%20at%2017.45.46-min,h_400,q_80,w_640.png)
Sine and cosine
Sine and cosine
- The tangent function is defined as the slope of the radius connecting the point and the origin:
![Illustrative background for Special coordinates](https://image-v2.cdn.app.senecalearning.com/2020-01/34206ccf-7252-43d3-8365-f8792ddda029/Screenshot%202020-01-24%20at%2017.45.42-min,h_400,q_80,w_640.png)
![Illustrative background for Special coordinates ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-01/34206ccf-7252-43d3-8365-f8792ddda029/Screenshot%202020-01-24%20at%2017.45.42-min,h_400,q_80,w_640.png)
Special coordinates
Special coordinates
- The point on the unit circle with coordinates corresponds to an angle of 90° or
.
- is undefined.
- By continuing around the circle, we can define the sine and cosine of all real values of this way as we go around the circle.
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
Jump to other topics
1Proof
1.1Types of Numbers
1.2Notation
2Algebra & Functions
2.1Powers & Roots
2.2Quadratic Equations
2.3Inequalities
2.4Polynomials
2.5Graphs
2.7Transformation of Graphs
3Coordinate Geometry
3.1Straight Lines
3.2Circles
3.2.1Equations of Circles centred at Origin
3.2.2Finding the Centre & Radius
3.2.3Equation of a Tangent
3.2.4Circle Theorems - Perpendicular Bisector
3.2.5Circle Theorems - Angle at the Centre
3.2.6Circle Theorems - Angle at a Semi-Circle
3.2.7Equation of a Perpendicular Bisector
3.2.8Equation of a Circumcircle
3.2.9Circumcircle of a Right-angled Triangle
3.3Parametric Equations (A2 only)
4Sequences & Series
4.1Binomial Expansion
5Trigonometry
5.2Trigonometric Functions
5.3Triangle Rules
6Exponentials & Logarithms
6.1Exponentials & Logarithms
7Differentiation
7.1Derivatives
7.2Graphs & Differentiation
7.3Differentiation With Trigonometry and Exponentials
7.4Rules of Differetiation (A2 only)
7.5Parametric & Implicit Differentiation
8Integration
8.1Integration
9Numerical Methods
9.1Finding Solutions
9.2Finding the Area
10Vectors
10.12D Vectors
10.23D Vectors
10.3Vector Proofs
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