Mathematics
Grade 12
15 min
Law of Cosines
Law of Cosines
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
State the three standard forms of the Law of Cosines and their rearranged versions for finding angles.
Derive the Law of Cosines using the distance formula and coordinate geometry.
Solve for an unknown side of an oblique triangle given two sides and the included angle (SAS).
Solve for any unknown angle of an oblique triangle given all three side lengths (SSS).
Determine when to apply the Law of Cosines versus the Law of Sines based on the given information.
Model and solve real-world problems involving vectors, navigation, and surveying using the Law of Cosines.
How can an architect calculate the length of a sloped roof beam without a right angle to work with? 📐 The Law of Cosines provides the answer!
This tutorial explores the Law of Cosines, a powerfu...
2
Key Concepts & Vocabulary
TermDefinitionExample
Oblique TriangleA triangle that does not have a right angle (90°). It can be either acute (all angles less than 90°) or obtuse (one angle greater than 90°).A triangle with angles 50°, 60°, and 70° is an acute oblique triangle. A triangle with angles 30°, 40°, and 110° is an obtuse oblique triangle.
Included AngleThe angle located between two specified sides of a triangle.In triangle ABC, angle C is the included angle between sides a and b.
SAS (Side-Angle-Side) CaseA scenario in trigonometry where the lengths of two sides and the measure of their included angle are known.Given side a = 10, side b = 12, and the included angle C = 45°. This is an SAS case, solvable by the Law of Cosines.
SSS (Side-Side-Side) CaseA scenario in trigonometry where the lengths of all three...
3
Core Formulas
Law of Cosines: Standard Form (Finding a Side)
c^2 = a^2 + b^2 - 2ab \cos(C)
Use this form when you know two sides (a and b) and their included angle (C) and you want to find the length of the third side (c). The formula can be written for any side: a² = b² + c² - 2bc cos(A) and b² = a² + c² - 2ac cos(B).
Law of Cosines: Rearranged Form (Finding an Angle)
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
Use this form when you know all three sides of a triangle (a, b, and c) and you want to find the measure of an angle (C). To find the angle, you must take the inverse cosine (arccos) of the result.
4 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
In a triangle, the sides are given by x, x+2, and x+4. The angle opposite the longest side is 120°. What is the value of x?
A.2
B.3
C.5
D.7
Challenging
The Law of Cosines is derived using the distance formula d = √((x₂-x₁)² + (y₂-y₁)²). If vertex C is at the origin (0,0), vertex B is at (a,0), and vertex A is at (b cos(C), b sin(C)), what expression results from applying the distance formula to find the length of side c (the distance between A and B)?
A.c² = (b cos(C) - a)² + (b sin(C) - 0)²
B.c² = (a cos(C) - b)² + (a sin(C) - 0)²
C.c² = (b cos(A) - a)² + (b sin(A) - 0)²
D.c² = (a - 0)² + (b - 0)²
Challenging
A triangular field has sides of 50m, 70m, and 80m. First, find the angle opposite the 70m side using the Law of Cosines. Then, calculate the area of the field using the formula Area = (1/2)ab sin(C). What is the approximate area?
A.1750 m²
B.1510 m²
C.1825 m²
D.1693 m²
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free