Questions: Law of Cosines

Question 1

A triangle has sides a = 5, b = 7, and included angle C = 120°. When you apply the Law of Cosines, the correction term −2ab·cos(C) will be:

• A. Negative, making c² smaller than a² + b²
• B. Zero, because cos(120°) = 0
• C. Positive, making c² larger than a² + b²
• D. Irrelevant — the Law of Cosines only applies when C is acute

Answer: (C)

When C is obtuse (greater than 90°), cos(C) is negative. So −2ab·cos(C) becomes −2ab·(negative number) = a positive value. This positive correction stretches c² above a² + b², which makes geometric sense: an obtuse angle splays the two sides apart, making the opposite side longer than the Pythagorean theorem would predict for a right triangle. This is one of the most commonly missed subtleties — students who forget the sign of cos(C) for obtuse angles will miscalculate.

Question 2

You know all three sides of a triangle (SSS) and want to find an angle. Which approach uses the Law of Cosines correctly?

• A. Solve c² = a² + b² − 2ab·cos(C) for c, then use the result to find C
• B. Rearrange to cos(C) = (a² + b² − c²) / (2ab), then apply arccosine
• C. Use the Law of Sines: sin(C)/c = sin(A)/a, since it's simpler for SSS
• D. SSS is indeterminate — you need at least one angle to apply the Law of Cosines

Answer: (B)

For SSS (all three sides known), rearrange the formula algebraically: cos(C) = (a² + b² − c²) / (2ab). Then C = arccos of that value. Option A solves for c (a side, not an angle), so it's wrong for this task. Option C is incorrect because Law of Sines requires at least one angle for SSS triangles; it is ambiguous and cannot be initiated from three sides alone. SSS is perfectly determinate — three side lengths uniquely define a triangle's angles.

Question 3

For an obtuse triangle where angle C > 90°, the side c opposite angle C is longer than what the Pythagorean theorem would predict.

• T. True
• F. False

Answer: True

The Pythagorean theorem predicts c² = a² + b² for a right angle. The correction term −2ab·cos(C) adjusts for the actual angle. When C > 90°, cos(C) < 0, so −2ab·cos(C) > 0, adding to a² + b². Therefore c² > a² + b², meaning c exceeds the Pythagorean prediction. This matches the geometric intuition: opening angle C beyond 90° pushes the vertex farther away, stretching the opposite side.

Question 4

When C = 90°, the Law of Cosines reduces to c² = a² + b² because the correction term −2ab·cos(C) equals zero.

• T. True
• F. False

Answer: True

cos(90°) = 0 exactly, so −2ab·cos(90°) = 0, and the formula reduces to the Pythagorean theorem. This is not a coincidence — the Law of Cosines is the generalization, and the Pythagorean theorem is the special case. Understanding this relationship shows that the Pythagorean theorem is not a separate law but a particular instance of a more general geometric truth about how side lengths and angles relate in any triangle.

Question 5

Why does the correction term −2ab·cos(C) in the Law of Cosines change sign depending on whether C is acute or obtuse, and what does each case mean geometrically?

Model answer: When C is acute, cos(C) > 0, so −2ab·cos(C) < 0 — c² is less than a² + b², meaning the opposite side is shorter than the right-triangle prediction. A narrow angle brings the two sides closer together. When C is obtuse, cos(C) < 0, so −2ab·cos(C) > 0 — c² exceeds a² + b², meaning the opposite side is longer. An obtuse angle splays the sides apart. The correction term adjusts the Pythagorean prediction based on how much the triangle deviates from a right angle.

The sign behavior of the correction term is the key to understanding the Law of Cosines as a continuous generalization of the Pythagorean theorem. At C = 90°, the correction is exactly zero. Below 90°, it subtracts; above 90°, it adds. This sign flip at 90° mirrors the behavior of cosine itself: positive in the first quadrant, zero at 90°, negative in the second. Recognizing this connection makes the formula memorable rather than arbitrary.