Law of sines
Use the law of sines to measure the lengths of a triangle
Autor:
Título 4
Learning objectives :
This activity allows students to measure inaccessible distances using the principles of trigonometry. It concretely applies the law of sines in a field context.
Concepts covered
Trigonometry; Law of sines; Theodolite; Triangulation; Indirect measures
What students will do :
The student uses the FizziQ theodolite to measure the angles between three fixed points in the playground. After directly measuring one of the distances of the triangle formed, the student applies the law of sines to calculate the other sides then creates a scale diagram and checks the consistency of his results.
What is required :
Smartphone with the FizziQ application; An open space with three visible reference points; A tape measure or a means of measuring distance; Drawing materials (protractor ruler paper); FizziQ experience notebook
Scientific background :
The law of sines is a fundamental theorem of trigonometry which states that in any triangle, the ratios between the lengths of the sides and the sines of the opposite angles are equal: a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the lengths of the sides and A, B, C are the opposite angles respectively. This relationship makes it possible to determine the dimensions of a triangle when certain of its elements are known, particularly in situations where certain direct measurements are impossible. The FizziQ digital theodolite uses the smartphone's orientation sensors (gyroscope and magnetometer) to measure azimuth, that is to say the horizontal angle between a direction and magnetic north. For each vertex of the triangle, the student measures the azimuth towards the other two points, then calculates the angle at the vertex by the difference of these two values. Once the three angles have been determined, simply measure one of the sides directly to be able to calculate the other two by applying the law of sines: if we know side a and all the angles, then b = a×sin(B)/sin(A) and c = a×sin(C)/sin(A). This triangulation technique is historically the basis of cartography and geodesy. Before the advent of GPS, this was the primary method for making accurate maps. It remains fundamental to understanding the principles of localization. Sources of error include the limited accuracy of the digital theodolite (±1-2°), inaccuracies in direct measurement of the reference side, and possible calculation errors. This activity perfectly illustrates the practical usefulness of trigonometry in solving concrete problems.