The Pythagorean Identity
Aligned to F-TF.C.8 — Common Core State Standards for Mathematics.
What this lesson teaches
A point on the unit circle is (cos θ, sin θ) and lies a distance 1 from the origin, so x^2 + y^2 = 1 becomes sin^2(θ) + cos^2(θ) = 1. Given one value and the quadrant, you can find the others.
Worked example
If sin(θ) = 3/5 and θ is in Quadrant I, then cos^2(θ) = 1 − 9/25 = 16/25, so cos(θ) = 4/5 (positive in Quadrant I). Then tan(θ) = sin(θ)/cos(θ) = (3/5)/(4/5) = 3/4.
Practice questions
- If cos(θ) = 0.6, find sin^2(θ).
- If cos(θ) = −12/13 and θ is in Quadrant II, find sin(θ).
- Given sin(θ) = √2/2 in Quadrant I, find cos(θ) and tan(θ).
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
La identidad pitagórica
Un punto del círculo unitario es (cos θ, sin θ) y está a distancia 1 del origen, así que x^2 + y^2 = 1 se convierte en sin^2(θ) + cos^2(θ) = 1. Dado un valor y el cuadrante, puedes hallar los demás.
Ejemplo: Si sin(θ) = 3/5 y θ está en el Cuadrante I, entonces cos^2(θ) = 1 − 9/25 = 16/25, así que cos(θ) = 4/5 (positivo en el Cuadrante I). Luego tan(θ) = sin(θ)/cos(θ) = (3/5)/(4/5) = 3/4.
More Functions lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required