5.4.3

Angle Addition Formulae

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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.

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Sine

  • We can write the sine of the sum of two angles AA and BB using the following identity:
    • sin(A+B)sinAcosB+cosAsinB\sin(A+B) \equiv \sin A\cos B + \cos A\sin B
  • We can write the cosine of the sum of two angles AA and BB using the following identity:
    • cos(A+B)cosAcosBsinAsinB\cos(A+B) \equiv \cos A\cos B -\sin A\sin B
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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 AA
    • The lower triangle (triangle 2) has a left corner angle equal to BB
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Label triangle 1

  • Using the fact that the hypotenuse is 1 and the left corner angle is AA, we can label the sides of the upper triangle using trigonometry:
    • Adjacent = hypotenuse ×cos(angle)=cosA\times\cos(\text{angle}) = \cos A
    • Opposite = hypotenuse ×sin(angle)=sinA\times\sin(\text{angle}) = \sin A
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Label triangle 2

  • Using the fact that the hypotenuse is cosA\cos A and the left corner angle is BB, we can label the sides of the lower triangle using trigonometry:
    • Adjacent = hypotenuse ×cos(angle)=cosAcosB\times\cos(\text{angle})= \cos A\cos B
    • Opposite = hypotenuse ×sin(angle)=cosAsinB\times\sin(\text{angle})=\cos A\sin B
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Triangle 3

  • Notice that the hypotenuse of triangle 1 forms another right-angled triangle with angle A+BA+B. By trigonometry:
    • Adjacent = cos(A+B)\cos(A+B)
    • Opposite = sin(A+B)\sin(A+B)
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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 BB.
    • This means the right corner angle of triangle 4 is equal to 90 B- B.
    • So the top corner angle is equal to BB.
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Find sides of triangle 4

  • The hypotenuse of triangle 4 is sinA\sin A. By trigonometry:
    • Opposite = hypotenuse ×cos(angle)=sinAcosB\times \cos(\text{angle}) = \sin A \cos B
    • Adjacent = hypotenuse ×sin(angle)=sinAsinB\times \sin(\text{angle}) = \sin A \sin B
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Find opposite side of triangle 3

  • The length of the opposite side of triangle 3 is equal to:
    • Opposite = sin(A+B)=sinAcosB+cosAsinB\sin(A+B) = \sin A \cos B + \cos A\sin B
  • Which is the angle addition formula for sine.
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Find adjecent side of triangle 3

  • The length of the adjacent side of triangle 3 is equal to:
    • Opposite = cos(A+B)=cosAcosBsinAsinB\cos(A+B) = \cos A \cos B - \sin A\sin B
  • Which is the angle addition formula for cosine.

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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