Unit 07 · Lesson 02 · Week 20 · Triangle Trigonometry, Bearings & Applications

Law of Sines and the Ambiguous Case

Solve right and oblique triangles with trigonometric ratios, Laws of Sines and Cosines, area formulas, bearings, and uncertainty analysis. In this lesson, the central focus is law of sines and the ambiguous case. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.

Open-AgePrintableFoundation · Core · ExtendedMastery target 80%

Learning intentions

Interpret the defining quantities and notation used in law of sines and the ambiguous case.
Construct or analyze at least two representations related to law of sines and the ambiguous case.
Show symbolic work with correct domains, units, intervals, or angle measures.
Explain why the selected method is valid and check the result against another representation.

Essential question

How do right-triangle ratios and oblique-triangle laws solve measurement, navigation, survey, and design problems?

Multiple-representation model

Solve right and oblique triangles with trigonometric ratios, Laws of Sines and Cosines, area formulas, bearings, and uncertainty analysis.

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}c^2=a^2+b^2-2ab\cos CK=\frac12ab\sin C

Worked reasoning

Oblique triangleGiven a=8, b=11, C=42°, use c²=a²+b²−2ab cos C, then use the Law of Sines to find a remaining angle.

PrecisionKeep full calculator precision through intermediate steps and round only the final measured result to the requested tolerance.

Misconception check: SSA may produce zero, one, or two triangles; a calculator angle alone does not settle the ambiguous case.

Communication and reasoning

Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.

Khaemenes Academy · Open-Age Precalculus

Law of Sines and the Ambiguous Case

Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.

U07 · L02

Name

Date

In a right triangle, opposite=6 and hypotenuse=10. Find sin θ.

Given a=7, A=35°, B=62°, find angle C.

Which law fits SAS data?

Find c when a=5, b=8, C=60°.

Find the area with a=10, b=12, C=30°.

A bearing of N35°E corresponds to what standard compass description?