5.4.3
Angle Addition Formulae
Angle Addition Formulae
Angle Addition Formulae
Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values.
Sine
Sine
- We can write the sine of the sum of two angles and
using the following identity:
- We can write the cosine of the sum of two angles and
using the following identity:
Proof
Proof
- Consider two right-angled triangles stacked on top of each other as shown.
- The upper triangle (triangle 1) has a hypotenuse equal to 1, with the left corner angle equal to
- The lower triangle (triangle 2) has a left corner angle equal to
Label triangle 1
Label triangle 1
- Using the fact that the hypotenuse is 1 and the left corner angle is , we can label the sides of the upper triangle using trigonometry:
- Adjacent = hypotenuse
- Opposite = hypotenuse
Label triangle 2
Label triangle 2
- Using the fact that the hypotenuse is and the left corner angle is
, we can label the sides of the lower triangle using trigonometry:
- Adjacent = hypotenuse
- Opposite = hypotenuse
Triangle 3
Triangle 3
- Notice that the hypotenuse of triangle 1 forms another right-angled triangle with angle . By trigonometry:
- Adjacent =
- Opposite =
Find angles of triangle 4
Find angles of triangle 4
- Triangle 4 is formed by the opposite side of triangle 1 as shown.
- By alternate angle theorem, the angle beneath triangle 4 is equal .
- This means the right corner angle of triangle 4 is equal to 90 .
- So the top corner angle is equal to .
Find sides of triangle 4
Find sides of triangle 4
- The hypotenuse of triangle 4 is . By trigonometry:
- Opposite = hypotenuse
- Adjacent = hypotenuse
Find opposite side of triangle 3
Find opposite side of triangle 3
- The length of the opposite side of triangle 3 is equal to:
- Opposite =
- Which is the angle addition formula for sine.
Find adjecent side of triangle 3
Find adjecent side of triangle 3
- The length of the adjacent side of triangle 3 is equal to:
- Opposite =
- Which is the angle addition formula for cosine.
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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