Pythagorean Identities

Trigonometry · Axiom Academy

Discover how the unit circle gives us three fundamental trigonometric identities through geometric insight and algebraic manipulation. On the unit circle (radius = 1), any point can be expressed as (cos θ, sin θ). Let's see why this immediately gives us our first Pythagorean identity. The distance from the origin to any point (x, y) on the circle is given by the Pythagorean theorem: x² + y² = r². Since r = 1 for the unit circle, and x = cos θ and y = sin θ, we get: 2. Deriving the Tangent-Secant Identity Starting with sin²x + cos²x = 1, we can derive a new identity by dividing every term by cos²x. Watch as we transform the fundamental identity: Recall that tan x = sin x / cos x and sec x = 1 / cos x. After dividing through by cos²x, each term simplifies to these familiar functions: 3. Deriving the Cotangent-Cosecant Identity Using the same approach, we can divide sin²x + cos²x = 1 by sin²x instead. This yields our third Pythagorean identity: Remember that cot x = cos x / sin x and csc x = 1 / sin x. Dividing by sin²x transforms each term accordingly: All three Pythagorean identities stem from the same geometric truth about the unit circle. Let's see them together: These identities are essential tools for simplifying trigonometric expressions, solving equations, and proving other trigonometric relationships. Memorize their forms, but more importantly, understand their origins!

This is the written version of the interactive lesson above. See the full Trigonometry course.