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 examples
Example 1
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.
Example 2
Find a missing angle using the Law of Sines (ASA situation, then a side). A triangle has angle A = 40°, angle B = 75°, and side a = 12. Then angle C = 180 - 40 - 75 = 65°. Use a/sin A = b/sin B: 12/sin 40° = b/sin 75°. Since sin 40° is about 0.643 and sin 75° about 0.966, b = 12 × 0.966/0.643 which is about 18.0.
Example 3
Common mistake: reaching for the Law of Sines when you are given two sides and the angle between them. With a = 8, b = 6 and C = 60°, the Law of Sines cannot start — every ratio it offers is missing either the side or the angle. SAS and SSS need the Law of COSINES; ASA, AAS and SSA need the Law of SINES. Check which three parts you actually hold before choosing.
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 1: 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.
Ejemplo 2: Halla un lado que falta con la ley de senos. Un triángulo tiene ángulo A = 40°, ángulo B = 75° y lado a = 12. Entonces el ángulo C = 180 - 40 - 75 = 65°. Usa a/sen A = b/sen B: 12/sen 40° = b/sen 75°. Como sen 40° es aproximadamente 0.643 y sen 75° aproximadamente 0.966, b = 12 × 0.966/0.643, unos 18.0.
Ejemplo 3: Error común: recurrir a la ley de senos cuando te dan dos lados y el ángulo entre ellos. Con a = 8, b = 6 y C = 60°, la ley de senos ni siquiera puede empezar — a cada razón que ofrece le falta o el lado o el ángulo. LAL y LLL necesitan la ley de COSENOS; ALA, AAL y LLA necesitan la ley de SENOS. Revisa qué tres partes tienes realmente antes de elegir.
More Similarity, Right Triangles & Trigonometry 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