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Three Towns

Part of the Triangles Without Right Angles section of Coddy's Trigonometry journey. Lesson 27 of 44.

Road distances between three towns give three sides and no angles. That can only be the law of cosines. No angle is known, so no side has a partner yet.

$Three towns X, Y and Z with distances 55, 60 and 70 km$

Towns 55, 60 and 70 km apart, and the angle φ at X between the 55 and 70 km roads: 60^2 = 55^2 + 70^2 - 2×55×70×cos(phi), so cos φ = 4325/7700 and φ is about 55.83°. Once one angle is known, the law of sines could finish the rest.

Challenge

Three towns P, Q and R are 42 km, 57 km and 68 km apart: PQ = 68, PR = 42 and QR = 57. Find the angle θ at P between the roads to Q and R, to two decimal places. Choose the law yourself, then press Check.

Try it yourself

$Use the figure to find theta$

Use the figure to find theta

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Triangles Without Right Angles

1Across the Lake

Across the Lake Sines Past 90°Cosines Past 90°Recap: Across the River

4Law of Cosines: SAS

The Law of Cosines An Obtuse Angle Between Through the Hill A Hooked Drive Recap: Two Sides, One Angle

7Area of a Triangle

Area With a Sine Area With an Obtuse Angle Working Backwards The Angle From the Area Heron's Formula Recap: A Garden Bed

2Law of Sines: AAS and ASA

Why the Law of Sines Holds Pair Side With Angle The Third Angle First Two Coast Stations Recap: Law of Sines

5Law of Cosines: SSS

Three Sides, One Angle The Largest Angle A Triangular Field Recap: Three Sides

8Bearings and Navigation

Reading Bearings Two Ships From Port Turning Off Course Finding a Course Angle Recap: Ships at Sea

3The Ambiguous Case

Finding an Angle Two Triangles The Height Test An Obtuse Given Angle Radio Coverage Recap: How Many Triangles?

6Which Law to Use

Sines or Cosines?Not BetweenThree TownsRecap: Pick the Law

9Final Challenges

Final: Across the Canyon Final: Every Triangle Final: A Plot of Land Final: Ship and Plane Final: The Largest Angle