The Law of Cosines is a rule for connecting the three side lengths of a triangle with one of its angles. It is especially useful when a triangle is not a right triangle, so the Pythagorean theorem does not directly apply. Students use it to solve triangles when they know two sides and the included angle, or when they know all three sides and need an angle.
It is a core tool in geometry, trigonometry, navigation, physics, and engineering.
Understanding Math: The Law of Cosines
The cosine term measures how much one side points in the same direction as another side. Imagine placing side a along the ground. Side b leaves one end at angle C.
Part of side b extends along the ground, while the rest rises away from it. That horizontal part changes the distance to the third corner. In a right triangle, the horizontal part has a simple relationship with the sides.
In a slanted triangle, cosine describes it. This is why the law contains a correction based on the angle, rather than only adding two squared lengths.
The sign of that correction tells an important story about the triangle. When the included angle is acute, cosine is positive. The correction is subtracted, so the opposite side is shorter than it would be in a right triangle with the same two nearby sides.
When the angle is right, the correction disappears. When the angle is obtuse, cosine is negative.
Subtracting a negative amount increases the result, so the opposite side becomes longer. This gives a quick way to judge whether a calculated length fits the shape before trusting a calculator result.
Careful labeling prevents many errors. Each lowercase side name must be opposite the matching uppercase angle name. Before putting values into the rule, sketch the triangle and mark the known information.
For a two-side, included-angle problem, the known angle must sit between the two known sides. Using a different angle creates a different triangle and produces the wrong length. For a three-side problem, isolate the cosine of the desired angle before using inverse cosine.
Keep the calculator in degree mode when the given angles use degrees. Round only near the end, since early rounding can noticeably change a final angle.
The law of sines depends on a known side and its opposite angle as a matched pair. Without that pair, it has no starting link. The cosine method fills that gap when information is given around a corner or when every side is known.
Students meet this idea in surveying, where two measured distances and the angle between them locate a point across a field. It appears in physics when two forces act at an angle and their combined effect must be found.
In computer graphics and engineering, angled supports, cables, and moving parts form triangles that are rarely right triangles. A final check should compare side sizes with angle sizes and confirm that all three angles would add to one hundred eighty degrees.
Key Facts
- Main formula: c^2 = a^2 + b^2 - 2ab cos(C)
- Equivalent forms: a^2 = b^2 + c^2 - 2bc cos(A) and b^2 = a^2 + c^2 - 2ac cos(B)
- Use SAS when two sides and the included angle are known, then solve for the opposite side.
- Use SSS by rearranging: cos(C) = (a^2 + b^2 - c^2) / (2ab)
- If C = 90°, then cos(90°) = 0, so c^2 = a^2 + b^2.
- The largest side is always opposite the largest angle, which helps check if an answer is reasonable.
Vocabulary
- Law of Cosines
- A formula that relates the three sides of any triangle to the cosine of one angle.
- Included angle
- The angle formed between two known sides of a triangle.
- SAS
- A triangle information pattern where two sides and the included angle are known.
- SSS
- A triangle information pattern where all three side lengths are known.
- Opposite side
- The side across from a given angle in a triangle.
Common Mistakes to Avoid
- Using the wrong angle in the formula. The angle in c^2 = a^2 + b^2 - 2ab cos(C) must be the angle between sides a and b and opposite side c.
- Forgetting the negative sign before 2ab cos(C). This changes the relationship and can make an obtuse triangle look incorrectly small.
- Applying the Pythagorean theorem to every triangle. a^2 + b^2 = c^2 only works for right triangles, while the Law of Cosines works for any triangle.
- Rounding too early in multi-step problems. Early rounding can shift the final side or angle by a noticeable amount, so keep extra decimal places until the end.
Practice Questions
- 1 A triangle has a = 7, b = 10, and C = 60°. Find side c using the Law of Cosines.
- 2 A triangle has sides a = 8, b = 11, and c = 13. Find angle C to the nearest degree.
- 3 Explain why the Law of Cosines becomes the Pythagorean theorem when the included angle is 90°.