The Law of Cosines

Textbook reading

The problem that opens the lesson

Two sides of a triangle are $11$ and $16$ with included angle $124$ degrees. Find the third side.

Solution

Begin by identifying the mathematical object and the information that fixes it. Identify the included angle or opposite side correctly, substitute with grouped products, solve, and check side-angle ordering. The relevant conditions are not optional bookkeeping: When solving for an angle, the computed cosine must lie in $[-1,1]$ . Rounding intermediate side values may distort later angles. Following that structure gives $c=sqrt(11^2+16^2-2(11)(16)cos124 degrees)\approx 22.41$ .

Why this works

SAS data determine the opposite side; SSS data determine angles. For SSS, finding the largest angle first provides a useful validity and rounding check. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

What this lesson is really about

The Law of Cosines generalizes the Pythagorean theorem to any triangle.

In $c^2=a^2+b^2-2ab$ cos C, the correction term accounts for the included angle. At $C=90$ degrees, cosine is zero and the Pythagorean theorem returns.

The point is not merely to reproduce a formula. A learner should be able to identify the quantities or geometric objects involved, explain why the relationship has its stated form, and recognize when the same idea appears in a graph, table, diagram, or model.

Textbook reading

Why the relationship works

SAS data determine the opposite side; SSS data determine angles. For SSS, finding the largest angle first provides a useful validity and rounding check.

Textbook reading

A reliable way to work

Identify the included angle or opposite side correctly, substitute with grouped products, solve, and check side-angle ordering.

When solving for an angle, the computed cosine must lie in $[-1,1]$ . Rounding intermediate side values may distort later angles.

After the symbolic work is complete, check the result. Depending on the lesson, this may mean substituting into an original equation, comparing coordinates, examining a graph, checking units, testing an interval, or confirming that every branch of a periodic solution has been included.

Textbook reading

What commonly goes wrong

A common error is pairing the included angle with the wrong opposite side or omitting the factor $2ab$ .

The repair is to return to the definition and identify the first step where the invalid solution stops describing the original mathematical object. Later algebra cannot rescue a first step that changed the domain, orientation, branch, or meaning of the problem.

Textbook reading

Worked examples

Worked example 1

Two sides of a triangle are $11$ and $16$ with included angle $124$ degrees. Find the third side.

Solution

Begin by identifying the mathematical object and the information that fixes it. Identify the included angle or opposite side correctly, substitute with grouped products, solve, and check side-angle ordering. The relevant conditions are not optional bookkeeping: When solving for an angle, the computed cosine must lie in $[-1,1]$ . Rounding intermediate side values may distort later angles. Following that structure gives $c=sqrt(11^2+16^2-2(11)(16)cos124 degrees)\approx 22.41$ .

Why this works

SAS data determine the opposite side; SSS data determine angles. For SSS, finding the largest angle first provides a useful validity and rounding check. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Transfer example

Problem

Derive the formula from coordinate geometry.

Worked development

Identify the included angle or opposite side correctly, substitute with grouped products, solve, and check side-angle ordering. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. In $c^2=a^2+b^2-2ab$ cos C, the correction term accounts for the included angle. At $C=90$ degrees, cosine is zero and the Pythagorean theorem returns. Then apply the conditions explicitly: When solving for an angle, the computed cosine must lie in $[-1,1]$ . Rounding intermediate side values may distort later angles. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

The law supports distance, structural geometry, navigation, and vector magnitude calculations.

Reasoning example

Problem

Solve an SSS triangle for its largest angle first.

Worked development

Identify the included angle or opposite side correctly, substitute with grouped products, solve, and check side-angle ordering. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. In $c^2=a^2+b^2-2ab$ cos C, the correction term accounts for the included angle. At $C=90$ degrees, cosine is zero and the Pythagorean theorem returns. Then apply the conditions explicitly: When solving for an angle, the computed cosine must lie in $[-1,1]$ . Rounding intermediate side values may distort later angles. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

The law supports distance, structural geometry, navigation, and vector magnitude calculations.

Worked example 4: quick check

Which angle should be found first in an SSS triangle and why?

Solution

Begin by identifying the mathematical object and the information that fixes it. Identify the included angle or opposite side correctly, substitute with grouped products, solve, and check side-angle ordering. The relevant conditions are not optional bookkeeping: When solving for an angle, the computed cosine must lie in $[-1,1]$ . Rounding intermediate side values may distort later angles. Following that structure gives The largest angle, opposite the largest side, to reduce ambiguity and check plausibility.

Why this works

SAS data determine the opposite side; SSS data determine angles. For SSS, finding the largest angle first provides a useful validity and rounding check. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

Application and interpretation

The law supports distance, structural geometry, navigation, and vector magnitude calculations.

A contextual answer must include units, a meaningful domain, and the assumptions that make the model plausible. An exact mathematical relationship should not be diluted into a decimal unless a measurement or comparison requires it.

Practice