7.7
The Law of Cosines
The Law of Cosines is a fundamental result in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It…
Consider a triangle with sides a, b, and c, where each angle is named after the vertex opposite its corresponding side.
The Law of Cosines states that the square of one side equals the sum of the squares of the other two sides, minus twice the product of those sides multiplied by the cosine of the included angle.
It applies when two sides and the included angle are known. It also applies when all three sides are given, and the included angle is found using the inverse of cosine.
When the included angle is a right angle, the triangle becomes a right triangle. The cosine of ninety degrees is zero. This removes the cosine term from the formula and simplifies it to the Pythagorean Theorem.
In land surveying, the Law of Cosines is used when one side of a marked triangle is inaccessible, such as when it lies across a water body. The other two sides and the included angle are measured using surveying equipment. Their values are then substituted into the Law of Cosines to calculate the length of the unknown side without direct measurement.
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Q1: When should you use the Law of Cosines instead of the Pythagorean Theorem?
Use the Law of Cosines for non-right triangles where the Pythagorean Theorem does not apply. The Law of Cosines relates side lengths to the cosine of an angle, generalizing the Pythagorean Theorem. When the included angle is 90 degrees, the cosine term becomes zero, and the formula simplifies to the Pythagorean Theorem, making it a special case of the Law of Cosines.
Q2: What information do you need to apply the Law of Cosines?
The Law of Cosines applies in two scenarios: when two sides and the included angle between them are known, or when all three sides are given and you need to find an included angle using the inverse of cosine. In either case, you can calculate the unknown side or angle by substituting the known values into the Law of Cosines formula.
Q3: How does the Law of Cosines relate to the Pythagorean Theorem?
The Law of Cosines is a generalization of the Pythagorean Theorem. When the included angle is a right angle, the cosine of 90 degrees equals zero, which removes the cosine term from the formula. This simplification yields the Pythagorean Theorem, showing that the Pythagorean Theorem is a special case of the Law of Cosines for right triangles.
Q4: Why is the Law of Cosines useful in land surveying?
In land surveying, one side of a marked triangle may be inaccessible, such as when it crosses a water body. Surveyors measure the other two sides and the included angle using surveying equipment, then substitute these values into the Law of Cosines to calculate the unknown side length without direct measurement, enabling precise distance calculations in complex geometries.
Q5: What does the Law of Cosines formula express mathematically?
The Law of Cosines states that the square of one side equals the sum of the squares of the other two sides, minus twice the product of those sides multiplied by the cosine of the included angle. For a triangle with sides a, b, c and angle C opposite side c, the formula is c² = a² + b² − 2ab cos(C).
Q6: How do you find an unknown angle using the Law of Cosines?
When all three sides of a triangle are known, rearrange the Law of Cosines formula to isolate the cosine of the unknown angle. Apply the inverse trigonometric function to find the angle measure. This method works for any triangle and allows you to determine all angles once the side lengths are given.
Q7: What real-world applications use the Law of Cosines beyond surveying?
The Law of Cosines is applied in astronomy to calculate distances between celestial bodies, in robotics to determine joint angles in arm mechanisms, and in physics to resolve forces acting at angles. These applications rely on the law's ability to handle non-right triangles where direct measurement is impractical or impossible.