Trigonometric ratios and functions satisfy many interesting properties. Let us find some of them. We will use a geometric structure to visualize the triangles involved.[br][br]Students work in pairs (A and B).[br][br]1) Student A selects a value for the acute angle in the green triangle, [math]\alpha[/math] (this value must be acute, otherwise, disfigures the geometric structure).[br]2) Student B selects a position for the orange point.[br]Observe that in the geometric structure:[br]- The acute angle [math]\alpha[/math] in the yellow triangle is always equal to the green value.[br]- All triangles are square.[br]- Hypotenuse in the red triangle is unitary.[br]- Control boxes show hints about equalities on lengths.[br]3) By turns, students A and B complete all missing symbolic informationin the spreadsheet (remember to write all information using [b]“quotation marks”[/b]). Use expressions such as: [math]sin\left(\alpha\right)[/math],  [math]cos\left(\alpha\right)[/math], [math]sin\left(\beta\right)[/math], [math]cos\left(\beta\right)[/math], [math]sin\left(\alpha+\beta\right)[/math], [math]cos\left(\alpha+\beta\right)[/math].[br]4) Students write down conclusions about the observed algebraic equalities.