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 examples
Example 1
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.
Example 2
If cos(θ) = −5/13 and θ is in Quadrant III, find sin(θ) and tan(θ). From sin²(θ) = 1 − 25/169 = 144/169, sin(θ) = ±12/13. Quadrant III makes sine negative, so sin(θ) = −12/13. Then tan(θ) = (−12/13)/(−5/13) = 12/5, positive — as tangent always is in Quadrant III.
Example 3
Common mistake: writing sin²θ as sin(θ²). The notation sin²θ means (sin θ)² — square the OUTPUT, not the angle. With θ = 30°, sin²(30°) = (0.5)² = 0.25, while sin(900°) is a completely different number. The identity is (sin θ)² + (cos θ)² = 1, and reading it any other way makes it false.
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 1: 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.
Ejemplo 2: Si cos(θ) = −5/13 y θ está en el tercer cuadrante, halla sen(θ) y tan(θ). De sen²(θ) = 1 − 25/169 = 144/169, sen(θ) = ±12/13. El tercer cuadrante hace negativo al seno, así que sen(θ) = −12/13. Entonces tan(θ) = (−12/13)/(−5/13) = 12/5, positiva — como siempre lo es la tangente en el tercer cuadrante.
Ejemplo 3: Error común: escribir sen²θ como sen(θ²). La notación sen²θ significa (sen θ)² — se eleva al cuadrado el RESULTADO, no el ángulo. Con θ = 30°, sen²(30°) = (0.5)² = 0.25, mientras que sen(900°) es un número completamente distinto. La identidad es (sen θ)² + (cos θ)² = 1, y leerla de otro modo la vuelve falsa.
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