Trigonometric identities
Some trig equations hold for every angle where both sides are defined. Which ones, where do they come from, and what are they good for?
Where you are 13 inside
What this area is about
An identity is an equation true at every input where both sides are defined, such as \sin^2 x + \cos^2 x = 1. Either side may then replace the other. That swap is how trig expressions are simplified, how new formulas are proved from old ones, and how equations are brought to a form that can be solved.
How these fit together
The idea. What an identity is separates an identity from an equation to be solved.
Basic identities. Reciprocal and quotient identities, cofunction identities and the Pythagorean identities.
Procedures. Simplifying trig expressions and verifying identities.
Formulas for combined angles. Cosine of a sum or difference comes first; sine and tangent of a sum follow from it. From those come double angle formulas, power reducing and half angle formulas, product to sum and sum to product formulas and combining a sine and a cosine into one wave.
Using them. Solving trigonometric equations finds every angle that works.
A path through it
What an identity is, then quotient identities, then the Pythagorean identities, which open simplifying and verifying. Cofunction identities need only the first lesson. Cosine of a sum needs the Pythagorean identities; sine of a sum needs cosine of a sum and the cofunctions. Product to sum needs the cosine formula; double angle and one wave need the sine formula; power reducing needs double angle. Trig equations come after the Pythagorean and double angle formulas.
Next, in calculus: derivatives of the trig functions and trigonometric integrals.