Compound Angle Formulae
What are the compound angle formulae?
- There are six compound angle formulae (also known as addition formulae), two each for sin, cos and tan:
- For sin the +/- sign on the left-hand side matches the one on the right-hand side
- sin(A+B)≡sinAcosB + cosAsinB
- sin(A-B)≡sinAcosB - cosAsinB
- For cos the +/- sign on the left-hand side is opposite to the one on the right-hand side
- cos(A+B)≡cosAcosB - sinAsinB
- cos(A-B)≡cosAcosB + sinAsinB
- For tan the +/- sign on the left-hand side matches the one in the numerator on the right-hand side, and is opposite to the one in the denominator
- The compound angle formulae can all the found in the formula booklet, you do not need to remember them
When are the compound angle formulae used?
- The compound angle formulae are particularly useful when find the values of trigonometric ratios without the use of a calculator
- For example to find the value of sin15° rewrite it as sin (45 – 30)° and then
- apply the compound formula for sin(A – B)
- use your knowledge of exact values to calculate the answer
- The compound angle formulae are also used…
- … to derive further multiple angle trig identities such as the double angle formulae
- … in trigonometric proof
- … to simplify complicated trigonometric equations before solving
How are the compound angle formulae for cosine proved?
- The proof for the compound angle identity cos (A – B ) = cos A cos B + sin A sin B can be seen by considering two coordinates on a unit circle, P (cos A, sin A) and Q (cos B, sin B )
- The angle between the positive x- axis and the point P is A
- The angle between the positive x- axis and the point Q is B
- The angle between P and Q is B – A
- Using the distance formula (Pythagoras) the distance PQ can be given as
- |PQ|2 = (cos A – cos B)2 + (sin A – sin B)2
- Using the cosine rule the distance PQ can be given as
- |PQ|2 = 12 + 12 -2(1)(1)cos(B – A) = 2 - 2 cos(B – A)
- Equating these two formulae, expanding and rearranging gives
- 2 - 2 cos(B – A) = cos2A + sin2A + cos2B + sin2B –2 cos A cos B - 2sin A sin B
- 2 - 2 cos(B – A) = 2 – 2(cos A cos B + sin A sin B )
- Therefore cos (B – A) = cos A cos B + sin A sin B
- Changing -A for A in this identity and rearranging proves the identity for cos (A + B)
- cos (B – (-A)) = cos(-A) cos B + sin(-A) sin B = cos A cos B – sin A sin B
How are the compound angle formulae for sine proved?
- The proof for the compound angle identity sin (A + B ) can be seen by using the above proof for cos (B – A) and
- Considering cos (π/2 – (A + B)) = cos (π/2)cos(A + B) + sin(π/2)sin(A + B)
- Therefore cos (π/2 – (A + B)) = sin(A + B)
- Rewriting cos (π/2 – (A + B)) as cos ((π/2 – A) + B) gives
- cos (π/2 – (A + B)) = cos (π/2 – A) cos B + sin (π/2 – A) sin B
- Using cos (π/2 – A) = sin A and sin (π/2 – A) = cos A and equating gives
- sin (A + B) = sin A cos B + cos A cos B
- Substituting B for -B proves the result for sin (A – B)
How are the compound angle formulae for tan proved?
- The proof for the compound angle identities tan (A ± B) can be seen by
- Rewriting tan (A ± B ) as
- Substituting the compound angle formulae in
- Dividing the numerator and denominator by cos A cos B
Exam Tip
- All these formulae are in the Topic 3: Geometry and Trigonometry section of the formula booklet – make sure that you use them correctly paying particular attention to any negative/positive signs
Worked Example
a)
Show that
b)
Hence, solve for