Addition and Double Angle Formulas

Addition Formulas

Addition Formulas allow us to express the sin, cos and tan of the sum or difference of two angles.
The formula for sin(a+b) is sin a cos b + cos a sin b. Substituting -b into this formula, we obtain the equation for sin(a-b).
The formula for cos(a+b) is cos a cos b - sin a sin b. The signs change when considering cos(a-b).
For tan(a+b) and tan(a-b), apply the sin and cos formulas to the fraction sin(a+b)/cos(a+b), simplify, and you will obtain their formulas.

Double angle formulas express trigonometric functions of twice an angle (2θ) in terms of functions of the original angle (θ).
The double angle formula for sin(2θ) is 2sin θ cos θ.
For cos(2θ), there are three forms: cos² θ - sin² θ, 2cos² θ - 1, and 1 - 2sin² θ. Note, all these forms are equivalent, use the version that is most relevant to your situation.
The double-angle formula for tan(2θ) is 2tan θ/(1 - tan² θ).

They are particularly useful for simplifying trigonometric expressions, solving trigonometric equations and proving identities.
The double-angle formulas allow us to reduce the power of a trigonometric function, which can be favourable in integral calculus.
Addition formulas can be used when dealing with wave-like phenomena in physics and engineering.

Committing these formulas to memory is useful, but understanding how to apply them is crucial.
Practise utilising these formulas with a variety of exercises. Start with those that require simple substitutions and work your way up to more complex identities and equations. Don’t forget to check your solutions!
It’s important to understand where these equations come from. If you are having difficulty remembering them, try proving them from the unit circle definition of trigonometric functions or via simple triangle geometry. This understanding will also come in handy if you ever forget the formulas in an assessment situation.