Law of sines

This activity allows students to measure inaccessible distances using the principles of trigonometry. It concretely applies the law of sines in a field context.

Long before satellites and GPS, explorers, surveyors, and military engineers measured distances across rivers, valleys, and mountain ranges using nothing more than a compass, a protractor, and the law of sines. This technique, called triangulation, was used by the ancient Greeks to estimate the size of the Earth, by Napoleonic surveyors to map France, and by the Great Trigonometric Survey to measure the height of Mount Everest. The principle is elegant: if you can measure one side and two angles of a triangle, the law of sines allows you to calculate all remaining sides and angles without ever physically crossing the distance. The law states that in any triangle, the ratio of each side length to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). Using FizziQ's built-in theodolite tool, which transforms the smartphone into an angle-measuring instrument, students can practice triangulation in their schoolyard and measure distances to inaccessible points, just as the great surveyors did centuries ago.

Activity overview:

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.

Level:

High school

FizziQ

Author:

Duration (minutes) :

40

What students will do :

- Measure angles between three fixed points using the FizziQ theodolite function
- Directly measure one side of the triangle and use the law of sines to calculate the others
- Verify the calculated distances by direct measurement when possible
- Understand the principle of triangulation and its historical applications
- Apply trigonometric relationships in a concrete, real-world context

Scientific concepts:

- Trigonometry
- Law of sines
- Theodolite
- Triangulation
- Indirect measures

Sensors:

- FizziQ theodolite (angle measurement using compass and gyroscope)

Material needed:

- 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

Experimental procedure:

Choose three clearly visible, fixed points in an open space (e.g., trees, posts, or buildings in a schoolyard). Label them A, B, and C.
Stand at point A and use the FizziQ Theodolite tool to measure the angle between the directions to points B and C. Record this angle as angle A.
Walk to point B and measure the angle between the directions to points A and C. Record this as angle B.
Walk to point C and measure the angle between the directions to points A and B. Record this as angle C.
Verify that the sum of the three angles is close to 180° (within ±5° is acceptable for smartphone measurements).
Using a tape measure, directly measure the length of one side of the triangle (e.g., side AB, the distance between points A and B). Record this as c (the side opposite angle C).
Apply the law of sines to calculate the other two sides: a = c × sin(A) / sin(C) and b = c × sin(B) / sin(C).
If accessible, directly measure sides a (BC) and b (AC) with the tape measure to verify your calculations.
Calculate the percentage error between the law-of-sines prediction and the direct measurement for each side.
Create a scale diagram of the triangle using a protractor and ruler. Verify that the diagram is geometrically consistent with your measurements.
Discuss the sources of error: angle measurement precision, alignment with the reference points, and the tape measure accuracy.
Try a second triangle with larger dimensions or with one side that is difficult to measure directly (e.g., across a pond or road), and use the law of sines to determine the inaccessible distance.

Expected results:

The FizziQ theodolite typically measures angles with a precision of ±2-3°, which translates to approximately 5-10% uncertainty in the calculated side lengths. For a triangle with sides of 20-50 meters, the law-of-sines calculation should agree with direct tape measurements within 10-15%. The sum of measured angles should be close to 180°, with deviations indicating measurement errors. Larger triangles (with sides over 30 m) tend to give better results because the relative angle measurement error is smaller. Students should find that the accuracy of the result is most sensitive to the angle measurements, particularly when angles are close to 0° or 180° (where the sine function changes slowly).

Scientific questions:

- Why is it sufficient to know one side and two angles to determine the entire triangle?
- Under what conditions does the law of sines fail or give ambiguous results?
- How did surveyors use triangulation to map entire countries before GPS existed?
- Why is the accuracy of the angle measurements more critical than the accuracy of the side measurement?
- How would you use triangulation to measure the width of a river without crossing it?
- What is the difference between the law of sines and the law of cosines, and when would you use each?

Scientific explanations:

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.

Extension activities:

Frequently asked questions:

Q: The sum of my three angles is not exactly 180°. Is my measurement wrong?
R: Small deviations (up to ±5°) are normal given the precision of smartphone angle measurements. If the sum is far from 180°, recheck your measurements. Magnetic interference from nearby metal objects can also affect the theodolite readings.

Q: How do I use the FizziQ theodolite accurately?
R: Hold the phone horizontally and point it precisely at the target. Wait for the reading to stabilize before recording. Avoid standing near large metal structures or electrical cables, which can distort the magnetic compass readings.

Q: One of my calculated side lengths is very different from the direct measurement. What went wrong?
R: The law of sines amplifies angle errors when angles are very small or very close to 180°. If one of the angles is less than 15° or greater than 165°, small measurement errors can cause large calculation errors. Try to choose triangle vertices that create angles between 30° and 120°.

Q: Can I use this technique to measure distances of several hundred meters?
R: Yes, triangulation works well at any scale. For very large triangles, the main limitation is the precision of the angle measurements, which remains ±2-3° regardless of scale. At 500 meters, a 2° error translates to about 17 meters of uncertainty.