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**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 value*s.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.

**Notes**

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.

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