# How Do You...Triangles

This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."

### Objective

Students define sin, cos, tan, csc, sec, and cot as the ratio of sides in a right triangle. Students identify certain special ratios for 30º, 45º, and 60º angles and organize them in the unit circle. Students can determine unknown sides and angles in a right triangle. Students should see that the size of an angle has a direct or inverse relationship to the value of certain side ratios.

### Progression

Other good resources on the topic come from Sam Shah (with a follow up) and Kate Nowak.

My approach and theirs have some similar messages. Students should understand that trig ratios mean something and that while 30/45/60 ratios are considered special, there's nothing particularly special about them other than reducing to nice-ish numbers.

First, I start with the "special" triangles. Have them draw a set of right triangles (4-5) of arbitrary size for 30/60/90, 60/30/90, and 45/45/90. Establish the definition of the opposite, adjacent, and hypotenuse. Measure the ratio of opposite/hypotenuse and adjacent/hypotenuse for your triangles and record them in a table.

They should see the ratios are the same for an angle regardless of triangle size. We use the three triangles and build a collection we call the Unit Circle where the hypotenuse is 1. I offer them things like root(2)/2 as the exact value of the ~0.71 number they see. We define opposite/hypotenuse as the sine of an angle, adjacent/hypotenuse as the cosine. It's important to take a moment and talk about opposite and adjacent as relative definitions. Some will catch that, others might not. They spend a week finding ratios on the unit circle and defining them as sin/cos/tan/csc/sec/cot.

Now we take a known angle and find the missing side of a right triangle. This is an exercise in vocabulary. Given the location of angle, what is the relative definition of the three sides? What trig ratio is relevant to the given information? This is the first time they compute sin/cos/tan values with a calculator. You could choose not to and save it for later.

Next, using a calculator I have them complete a table of values for sin/cos/tan of the angles 0 to 90 in increments of 5 (this is similar to Sam's large packet of triangles to build a database of ratio values). I give them a set of right triangles of arbitrary size and they use their table of ratios to approximate the angles. I avoid using the inverse trig functions on the calculator until we start talking about trig equations later on. I want them to make a connection between ratio value and angle value. I want them to have an idea of sin and cos trends to help when we graph those functions later.

Lastly, they create a set of right triangles, one in each quadrant, of arbitrary size where they demonstrate the values of the six trig functions for one interior angle and use those ratios to approximate the value of the interior angle.

They have an assignment with a set of right triangles where sides and angles are missing and they have to find the missing information using any of the methods we discussed.

### Activities

Most of their work through this unit is a series of activities. Here's a picture of the arbitrary special right triangles and the table of ratios:

Here's a picture of the ratio table they complete with the calculator:

Here's a picture of the final assignment, an arbitrary set of right triangles with all ratios computed and angles approximated:

Another option is the classic "how tall is the tall thing?" activity. Take them somewhere where you have some tall objects. Have them measure 1 side and 1 angle. Use a relevant trig ratio to approximate the height of the object and compare results.

### Assessment

Triangles cover 5 topics in my SBG system. Student demonstrate knowledge similar to classwork and in a couple cases are asked to draw triangles oriented correctly in a quadrant, label sides based on a defined trig ratio, and determine the rest of the information.

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