Explain the central structure of law of cosines & heron-type area reasoning using precise mathematical language.
Apply the governing relationship and verify results with an alternate representation or substitution.
State domain, unit, precision, and context restrictions whenever they affect the result.
Analyze a misconception and communicate a correction with evidence.
Essential question: How do identities and triangle laws transform expressions, solve periodic equations, and determine unknown measurements?
Governing relationship
c²=a²+b²−2ab cosC.
The Law of Cosines generalizes the Pythagorean theorem and solves SAS or SSS triangles. It also supports classification and distance models.
Worked examples
Choose the formula from the information given. SAS means two sides and their included angle; SSS means all three sides.
Investigation A · SAS: a=5, b=8, included angle C=60°.
Write the Law of Cosines for the side opposite C: c²=a²+b²−2ab cos C.
Substitute: c²=25+64−(2×5×8)cos60°=89−40=49, so c=7 (take the positive side length).
Find the area using the included angle: K=½ab sin C=½×5×8×sin60°=10√3≈17.32 square units.
Independently check with Heron's formula: s=(5+8+7)/2=10; K=√[10(10−5)(10−8)(10−7)]=√300=10√3. Agreement confirms the model.
Cross-check: 5²+7²>8², so this triangle is acute at the angle opposite side 8, consistent with the included 60° and side lengths.
Investigation B · SSS: a=3, b=4, c=6. Find angle C and area.
Check the triangle inequality: 3+4>6, so a nondegenerate triangle exists.
Rearrange the Law of Cosines: cos C=(a²+b²−c²)/(2ab)=(9+16−36)/24=−11/24.
Therefore C=arccos(−11/24)≈117.28°, which is obtuse. Side c=6 is the longest, so its opposite angle C must be the largest.
Heron's semiperimeter is s=(3+4+6)/2=6.5. Thus K=√[6.5×3.5×2.5×0.5]≈5.33 square units.
Verify using ½ab sin C≈½×3×4×sin117.28°≈5.33.
Visual model · included angles and opposite sides
Set the labels before substituting into c²=a²+b²−2ab cos C. Heron's formula uses the resulting three side lengths, independent of which side you choose as a base.
Guided and independent practice
Explain why the Law of Cosines reduces to the Pythagorean theorem when C=90°.
Using a=7, b=9, C=60°, solve for c and the included-angle area. Carry exact radicals before rounding.
For sides 5, 5, 6, use Heron's formula to determine the area and then check with an altitude.
Explain why a=2, b=3, c=6 cannot be used in Heron's formula for a real triangle.
Verify your model in Triangle Law Expedition; record which quantities were SAS or SSS.
Independent transfer · check only after solving
When a=3, b=4 and C=90°, find c and the area, then verify using Heron.
For side lengths 5, 5, 6, find s and K exactly.
For a=7, b=9, C=60°, express c in radical form.
Reveal worked verification and answers
c=√(9+16)=5, K=½×3×4=6; s=6 and Heron K=√[6×3×2×1]=6.
s=8 and K=√[8×3×3×2]=√144=12 square units.
c²=49+81−2×7×9×½=67, so c=√67≈8.19.
Differentiated pathways
Foundation
Review prerequisite vocabulary, annotate each worked step, complete the first four prompts, and use a table or graph to support symbolic work.
Core
Complete every prompt, explain domain and verification, correct errors, and submit the applied task with clear mathematical communication.
Extended
Generalize the relationship, compare methods, use technology strategically, analyze sensitivity, and create a proof, counterexample, or advanced model.
Application laboratory
Determine inaccessible distances and check measurement sensitivity.
Required evidence: defined quantities and units; at least two representations; transparent calculations; verification; one limitation; and a defensible conclusion.
Misconception and correction
Common misconception: Using the wrong side as opposite the named angle.
Write the first invalid step, the governing rule that corrects it, and one counterexample or verification that demonstrates the correction.
Mastery evidence
Accurate symbolic work with preserved equivalence.
Correct domain, interval, unit, and precision statements.
At least one independent verification.
A correction explaining why an incorrect approach fails.
A clear application conclusion that distinguishes mathematical result from recommendation.
Reflection
What structure did you recognize? Which representation was most useful? What assumption or restriction mattered most? What will you do differently on the next problem?