Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles
INTERPRETATION OF OBJECTIVE - G.SRT.C.6
We are now connecting similarity to right triangles where two angles are known, a right angle and another angle. This of course locks in the proportionality of the sides and we are able to establish ratios of the sides that hold for all triangles with those two angles - thus the link to trigonometry.
(1) The student will be able to label a triangle in relation to the reference angle (opposite, adjacent & hypotenuse).
(2) The student will be able to determine the most appropriate trigonometric ratio (sine, cosine, and tangent) to use for a given problem based on the information provided.
(3) The student will be able to solve for sides and angles of right triangles using trigonometry.
THE BIG IDEA
That similarity connects angles to sides and sides to angle and that we refer to that as Trigonometry. When we work in right triangles and we know one of the complementary angles we have similar triangles and the ratios of the sides are fixed. This allows us in a significant way move from side lengths to angles and vice versa.
TRAPS & PITFALLS
I don't know of any traps here... I would just say build all three ratios at the same time. Some textbooks of the past broke this up over a few days, I think students should learn them all at once and begin the decision making process about what has been provided and which ratio will help them solve the problem.
Trigonometry is connected heavily to similarity. The entire essence of trigonometry is the fact that the ratios of sine, cosine and tangent are constant for each reference angle... and the reason they are constant is because those triangles are all similar by AA.
These right triangle relationships allow us to build a formula for area and solve for values of oblique (non-right) triangle.
MY REFLECTIONS (over line l)
I love the connection from similarity to trigonometry. Using G.CO.B.6 Activity #1 has been a huge success in my classroom for many years. When we start with similarity and then build to determining the ratios of the sides and then finally looking them up on a trigonometry table.... THE LIGHTS GO ON!! Immediately, students understand what that value is that appears magically in their calculators. This starts demystifies the scarey trigonometry - students walk out of class saying... "I thought trigonometry was going to be difficult but I guess it just sounded difficult."