Applying the Laws of Sines and Cosines
Aligned to G-SRT.D.11 — Common Core State Standards for Mathematics.
What this lesson teaches
Use the Law of Sines when you know an angle and its opposite side plus one more piece (AAS or ASA). Use the Law of Cosines when you know two sides and the included angle (SAS) or all three sides (SSS). These solve triangles that are not right triangles.
Worked example
Find a side using the Law of Cosines (SAS). A triangle has a = 8, b = 6, and included angle C = 60°. Then c^2 = a^2 + b^2 - 2ab cos C = 64 + 36 - 2(8)(6)(0.5) = 100 - 48 = 52. So c = sqrt(52) which is about 7.2 units. The included angle is essential to pin down the third side.
Practice questions
- A triangle has angle A = 40°, angle B = 80°, and side a = 9. Find side b using the Law of Sines.
- Two sides of a triangle are 11 and 13 with included angle 100°. Find the third side with the Law of Cosines.
- A triangle has all three sides 6, 8, 10. Use the Law of Cosines to find the largest angle.
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
Aplicar las Leyes de Senos y Cosenos
Usa la Ley de Senos cuando conoces un ángulo y su lado opuesto más otro dato (AAS o ALA). Usa la Ley de Cosenos cuando conoces dos lados y el ángulo incluido (LAL) o los tres lados (LLL). Estas resuelven triángulos que no son rectángulos.
Ejemplo: Halla un lado con la Ley de Cosenos (LAL). Un triángulo tiene a = 8, b = 6 y ángulo incluido C = 60°. Entonces c^2 = a^2 + b^2 - 2ab cos C = 64 + 36 - 2(8)(6)(0.5) = 100 - 48 = 52. Así, c = raíz(52), que es aproximadamente 7.2 unidades. El ángulo incluido es esencial para fijar el tercer lado.
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