5.4.4
Manipulating the Angle Addition Formulae
Manipulating the Angle Addition Formulae
Manipulating the Angle Addition Formulae
We can prove other angle addition formulae by manipulating what we have already for and .
![Illustrative background for Sine subtraction formula](https://image-v2.cdn.app.senecalearning.com/2020-06/f5cf46e2-512a-4b52-8142-a8d96d33ab29/Screenshot%202020-06-18%20at%2014.19.09-min,h_400,q_80,w_640.png)
![Illustrative background for Sine subtraction formula ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/f5cf46e2-512a-4b52-8142-a8d96d33ab29/Screenshot%202020-06-18%20at%2014.19.09-min,h_400,q_80,w_640.png)
Sine subtraction formula
Sine subtraction formula
- We can find the identity for by substituting
into the identity for
:
- Where we have used and .
![Illustrative background for Cosine subtraction formula](https://image-v2.cdn.app.senecalearning.com/2020-06/3f672cfa-7e3f-4c33-9094-57cc608152ca/Screenshot%202020-06-18%20at%2014.19.10-min,h_400,q_80,w_640.png)
![Illustrative background for Cosine subtraction formula ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/3f672cfa-7e3f-4c33-9094-57cc608152ca/Screenshot%202020-06-18%20at%2014.19.10-min,h_400,q_80,w_640.png)
Cosine subtraction formula
Cosine subtraction formula
- We can find the identity for by substituting
into the identity for
:
- Where we have used and .
![Illustrative background for Tangent addition formula](https://image-v2.cdn.app.senecalearning.com/2020-06/fdf97c6c-7143-4b91-92dc-4ffbee4af924/Screenshot%202020-06-18%20at%2014.19.12-min,h_400,q_80,w_640.png)
![Illustrative background for Tangent addition formula ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/fdf97c6c-7143-4b91-92dc-4ffbee4af924/Screenshot%202020-06-18%20at%2014.19.12-min,h_400,q_80,w_640.png)
Tangent addition formula
Tangent addition formula
- To find the addition formula for tangent, we
- Substituting in our angle addition formulas, we get:
- Dividing both the numerator and denominator by a factor of gives:
![Illustrative background for content](https://image-v2.cdn.app.senecalearning.com/2020-06/912f980b-80a2-4330-a6e5-0b5b624fa988/Screenshot%202020-06-18%20at%2014.19.13-min,h_400,q_80,w_640.png)
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![Illustrative background for Tangent addition formula cont.](https://image-v2.cdn.app.senecalearning.com/2020-06/3e3f57f8-6e98-476c-9a17-42f017c05f61/Screenshot%202020-06-18%20at%2014.19.14-min,h_400,q_80,w_640.png)
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Tangent addition formula cont.
Tangent addition formula cont.
- Cancelling out common factors gives:
- Finally, we can simplify this as:
![Illustrative background for Negative tangent](https://image-v2.cdn.app.senecalearning.com/2020-06/a83219f9-c2eb-44d4-a547-300e1f456d90/Screenshot%202020-06-18%20at%2014.19.16-min,h_400,q_80,w_640.png)
![Illustrative background for Negative tangent ?? "content](https://image-v2.cdn.app.senecalearning.com/2020-06/a83219f9-c2eb-44d4-a547-300e1f456d90/Screenshot%202020-06-18%20at%2014.19.16-min,h_400,q_80,w_640.png)
Negative tangent
Negative tangent
- We can find the identity for by substituting
into the identity for
:
- Where we have used .
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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