Proving the Laws of Sines and Cosines
Aligned to G-SRT.D.10 — Common Core State Standards for Mathematics.
What this lesson teaches
The Law of Sines says a/sin A = b/sin B = c/sin C for any triangle, proved by dropping an altitude and expressing it two ways. The Law of Cosines, c^2 = a^2 + b^2 - 2ab cos C, generalizes the Pythagorean theorem to any angle C.
Worked example
Prove the Law of Sines with an altitude. In triangle ABC, drop altitude h from vertex C to side AB. In the two right triangles formed, sin A = h/b and sin B = h/a. Solve each for h: h = b sin A and h = a sin B. Setting them equal, b sin A = a sin B, which rearranges to a/sin A = b/sin B.
Practice questions
- In a triangle, a = 10, angle A = 30°, angle B = 45°. Use the Law of Sines to find b.
- Use the Law of Cosines to find side c when a = 5, b = 7, and angle C = 60°.
- Explain how the Law of Cosines reduces to the Pythagorean theorem when angle C = 90°.
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
Demostrar las Leyes de Senos y Cosenos
La Ley de Senos dice a/sen A = b/sen B = c/sen C para cualquier triángulo, demostrada trazando una altura y expresándola de dos formas. La Ley de Cosenos, c^2 = a^2 + b^2 - 2ab cos C, generaliza el teorema de Pitágoras a cualquier ángulo C.
Ejemplo: Demuestra la Ley de Senos con una altura. En el triángulo ABC, traza la altura h desde el vértice C al lado AB. En los dos triángulos rectángulos formados, sen A = h/b y sen B = h/a. Despeja h en cada uno: h = b sen A y h = a sen B. Igualándolos, b sen A = a sen B, que se reordena a a/sen A = b/sen B.
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