Seneca
Sign up

Angle Addition Formulae

Maths: Edexcel A Level Pure Maths, topic 5.4.3

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 AA and BB using the following identity:
  • sin⁡(A+B)≡sin⁡Acos⁡B+cos⁡Asin⁡B\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)≡cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B) \equiv \cos A\cos B -\sin A\sin B
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 AA
  • The lower triangle (triangle 2) has a left corner angle equal to BB
Label triangle 1

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)=cos⁡A\times\cos(\text{angle}) = \cos A
  • Opposite = hypotenuse ×sin⁡(angle)=sin⁡A\times\sin(\text{angle}) = \sin A
Label triangle 2

Label triangle 2

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

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)
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 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.
Find sides of triangle 4

Find sides of triangle 4

  • The hypotenuse of triangle 4 is sin⁡A\sin A. By trigonometry:
  • Opposite = hypotenuse ×cos⁡(angle)=sin⁡Acos⁡B\times \cos(\text{angle}) = \sin A \cos B
  • Adjacent = hypotenuse ×sin⁡(angle)=sin⁡Asin⁡B\times \sin(\text{angle}) = \sin A \sin B
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 = sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B) = \sin A \cos B + \cos A\sin B
  • 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 = cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B) = \cos A \cos B - \sin A\sin B
  • Which is the angle addition formula for cosine.

Test yourself on Angle Addition Formulae

  • What is the following expression equivalent to?
    sin⁡45cos⁡60+sin⁡60cos⁡45\sin45\cos60+\sin60\cos45

  • What is the following expression equivalent to?
    cos⁡116cos⁡25−sin⁡116sin⁡25\cos116\cos25-\sin116\sin25

  • What is the following expression equivalent to?
    cos⁡9cos⁡75−sin⁡9sin⁡75\cos9\cos75-\sin9\sin75

  • What is the following expression equivalent to?
    sin⁡32cos⁡51+cos⁡32sin⁡51\sin32\cos51+\cos32\sin51

Practise this topic for free