Thought for the day: Simple images can take you in many different mathematical directions.
Concepts: interior angles in triangles; the inverse tangent function; the tangent addition formula; predicting and verifying trigonometric identities (possibly radian measure, area, and/or the Pythagorean Theorem)
Examples of noticing and wondering
I notice three right triangles joined along one of each of their angles.
I notice three acute angles that have a sum of 180° (or π radians).
I notice that the total area of the triangles is 4 square units.
I notice that the squares of the sides of the middle triangle satisfy the equation 2 + 8 = 10.
I notice that the three colored angles illustrate the identity
arctan(1) + arctan(2) + arctan(3) = π.
I notice that the three angles in the upper left corner illustrate an identity involving π/2:
arctan(1) + arctan(1/2) + arctan(1/3) = π/2.
I wonder if a, b, and c can take on negative values.
I wonder if the reciprocals of three values that satisfy the first equation will always satisfy the second.
i wonder how difficult it is to prove this identity using standard trig identities.
I wonder if I can redraw the picture so that the three angles occur in a different order.
I wonder if I can discover other triples of numbers that satisfy the identify.
I wonder if there exist other whole numbers a, b, and c that satisfy the same identity.
I wonder if I can discover general relationships between a, b, and c such that
arctan(a) + arctan(b) + arctan(c) = π.
Students may verify the first identity using the definition of the arctangent function and the addition formula for the tangent (twice).
It is possible to draw this picture with the angles in different orders. It is also possible to use similar types of drawings to discover other combinations of a, b, and c that satisfy the identity and to develop algebraic expressions that describe the relationships between the three variables. it is interesting to try to discover these relationships both pictorially and algebraically. There are no other solutions in which all three values are natural numbers.