Thought for the day: "Proofs Without Words" often make excellent Creative Math Prompts, especially if you leave out details that students can fill in for themselves.
Concepts: addition formulas for trig functions, right triangle trigonometry, similarity
Examples of noticing and wondering
I notice a right triangle inscribed within a rectangle.
I notice that very few measurements are given.
I notice that expressions for the sides of the right triangle on the bottom are determined by the given information.
I notice that the right triangle on the bottom and the right triangle in the upper-right corner of the rectangle are similar.
I notice that the hypotenuses of these two triangles set the scale factor between them, which determines expressions for each side of the triangle on the upper right.
I notice that the left triangle shows an expression for the tangent of the sum of two acute angles.
I wonder if it is possible to determine expressions for all lengths and angles in the prompt.
I wonder if I would get the same expression for the tangent of the sum of alpha and beta if the length of the bottom side were not equal to 1.
I wonder what would happen if I set some other side length equal to 1.
I wonder if I could draw a picture to generalize my formula to angles that are not acute.
I wonder if I could use this picture (or create a new one) to prove a formula for the tangent of a difference of angles.
I wonder if I could use the same picture or create new pictures to prove addition formulas for sines and cosines.
Creating something new
As often happens, ideas for creating something new may emerge from students' "wonderings." For example, they may:
Change the side length on the bottom and recalculate the formula.
Try to use the picture to create a formula for the tangent of a difference of angles.
Create a new picture that shows the tangent of a difference.
Create analogous formulas for sine and cosines using this pictures or other pictures that they design.
Create other drawings or more general arguments to handle angles that are not acute.
Use the middle triangle and the left triangle to create multiple expressions for their common hypotenuse. Prove (independently) that the expressions are equivalent.
The following pictures show a possible thinking process for finding unknown angles and lengths and developing a formula for the tangent of a sum. You can find addition formulas for sine and cosine from the same prompt!
You can find expressions for the unknown angles using knowledge of interior angles in triangles, complementary and supplementary angles, alternate interior angles, etc. You can check your results by applying more than one method to a given angle.
The right-triangle definitions of trig functions make it possible to find expressions for the two unknown sides of the triangle on the bottom.
I chose to let x = sec(alpha) in order to make the remaining diagrams easier to read. In the middle triangle, the definition of the tangent makes it possible to find an expression for the side opposite the angle, beta. I am omitting an expression for the hypotenuse. Students will be able to calculate it in at least two ways by the time they have found expressions for the remaining lengths.
A quick glance at the angles shows that the two shaded triangles are similar. The previous picture shows that the scale factor between the triangles is tan(beta). These observations enable you to find expressions for the sides of the shaded triangle on top.
You may find expressions for the legs of the triangle on the left by observing that the opposite sides of the enclosing rectangle are congruent. I have shown one of the angles in this triangle again in order to prepare for a conclusion about the tangent of the sum of two angles.
By applying the right-triangle definition of the tangent function to the triangle on the upper-left, you obtain the result: