Unit 07 · Lesson 05 · Week 21 · Triangle Trigonometry, Bearings & Applications

Bearings, Navigation & Triangulation

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 bearings, navigation & triangulation. 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 bearings, navigation & triangulation.
Construct or analyze at least two representations related to bearings, navigation & triangulation.
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

Bearings, Navigation & Triangulation

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

U07 · L05

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?