Pythagorean identities · derivation ·proof
Where sin²θ + cos²θ = 1 comes from
From the definition of sine and cosine. You do not need to memorise this one — it is the Pythagorean theorem written in trigonometric notation. If you ever feel it is 'magic', the radius of the circle is the missing ingredient.
sin²θ + cos²θ = 1
Building it step by step
- x² + y² = r²Equation of a circle centred at the origin.
- x = cos θ, y = sin θ, r = 1Unit-circle definitions.
- cos²θ + sin²θ = 1Substitute and simplify.
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from sin(x)**2 + cos(x)**2 and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the pythagorean identity page.