There are four sets of formulae which make the solving and manipulation of trigonometric equations and identities easier.
All of these formulae are all provided in the tables issued for most examinations, although, as always, it is useful if you can memorise them.
The formulae and proofs are listed below, along with examples of their use.
Compound Angle Formulae
Trigonometric expressions such as sin (45 + 60) do not expand in a similar way to algebraic expansions.
i.e. sin (45 + 60) ≠ sin 45 + sin 60, instead the following formulae apply.
(Two proofs are given, the other formulae can be proven in a similar manner.)
sin(A + B) = sin A cos B + cos A sin B
sin(A − B) = sin A cos B − cos A sin B
cos(A + B)= cos A cos B − sin A sin B
cos(A − B) = cos A cos B + sin A sin B proof
Note the "reversal" of signs in the cosine formulae.
Simplify cos (180 − A)
cos (180 − A) = cos 180.cos A + sin 180. sin A = -1.cos A + 0.sin A = -cos A
Double Angle Formulae
A useful simplification of the compound angle formulae is when A and B are equal,
i.e. A + B = A + A = 2A
Again, trig functions do not obey algebraic rules, sin 2A ≠ 2sin A
By substituting into the compound angle formulae:
sin 2A = 2sin A cos A
cos 2A = cos2A − sin2A
cos 2A = 2cos 2A -1
cos 2A = 1 − sin2A
The three versions of the cosine double angle formulae are obtained using the
identity sin2 x + cos2 x = 1
If sin x = 0.6 and cos x = 0.8, find the value of sin 2x
sin 2x = 2sin x cos x = 2 x 0.6 x 0.8 = 0.96
Product To Sum Formulae
These formulae are useful when the product of two trig functions need to be changed to a sum.
2sin A cos B = sin (A + B) + sin (A − B) proof
2cos A sin B = sin (A + B) − sin (A − B) proof
2cos A cos B = cos (A + B) + cos (A − B) proof
2sin A sin B = cos (A − B) − cos (A + B) proof
Express the product 2sin 40° cos 30° as a sum of two trig. functions.
2 sin 40° cos 30° = sin (40 + 30)° + sin (40 − 30)° = sin 70° + sin 10°
Sum to Product Formulae
The sum of two trig functions is not obtained by adding the two angles together.
e.g. sin 40° + sin 30° ≠ sin 70°
These formulae are needed when the sum of two trig functions is changed to a product.
Express the sum sin 40° + sin 30° as the product of two trig. functions.
sin 40° + sin 30° = 2 sin (40 + 30)°/ 2 cos (40 − 30)°/ 2 = 2 sin 35° cos 5°