Reflection: Developing a Conceptual Understanding Definitions of the Trig Ratios and a Fascinating Chart - Section 3: Mini-Lesson: Using the trig ratios to figure out if you're right

 

If this is a property and that is a property, then maybe this and that have a relationship...

The trigonometric ratios are a property of similar triangles.  If both angles and ratios are properties of similar triangles, then those two properties have a relationship. These relationships are given the names sine, cosine, and tangent.  Students have seen the right triangle definitions for sine, cosine, and tangent before, so it feels like a review when we take a look at them.  

This activity dispels some of the mystery that surrounds the trigonometric ratios for many of my students.  Up until now, if a student typed sin(25) into their calculator, they would get some this sort of random looking decimal number: 0.422618... - Now, however, we've calculated the ratios first, and only after that have we brought in the angle measurement and word sine.  To give my students another way to think about it, I refer to these ratios as percentages.  For example, if there's a 25 degree angle, I point the opposite leg and the hypotenuse and ask, "Is it reasonable to say that this leg would be approximately 42% of the length of the hypotenuse?"  I continue, "if this angle were to get bigger, wouldn't this leg grow to a length greater than 42% of the length of the hypotenuse?"  When we're stating these ratios, we're talking about the relationships between the sides of the triangle, and these relationships are intimately connected to the measures of the angles.

That's what trig ratios are all about.

  How I intend students to be making sense of this...
  Developing a Conceptual Understanding: How I intend students to be making sense of this...
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Definitions of the Trig Ratios and a Fascinating Chart

Unit 5: Trigonometry: Triangles
Lesson 4 of 9

Objective: SWBAT use the definitions of the trigonometric ratios to check their measurements of a rectangle.

Big Idea: If similar triangles have congruent angles and proportional sides, then what is the relationship between these two properties?

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