Not really; if they know the sin is the y coordinate of the unit circle, just visualizing or drawing the unit circle will give the information sine starts at 1. If they can’t do this, I would say they don’t understand trig well enough to handle identities.

So yes, they have to memorize at some level, but it is the arbitrary convention of sin = opp / hyp, cos = adj / hyp, tan = opp / adj. Everything else can be pulled straight off the unit circle.

Tangent is sine divided by cosine because it is opposite over adjacent, that is, the y value divided by the x value on the unit circle. And yes, I would expect them to be able to visualize that, because otherwise they get lost in the different varieties of Pythagorean identities.

There is an alternate way to draw the unit circle which directly has tangent as one of the sides. I only use it if students seem to be really interested in the geometric aspect.

https://numberwarrior.wordpress.com/2008/03/27/trig-identities-and-the-unit-circle/

And it is possible to get kids to think about it. I do it all the time. Here’s an early version: https://educationrealist.files.wordpress.com/2012/03/trigquestion.jpg

I often give a triangle with clearly distinguishable leg lengths (but no measurements) and ask them to order the six ratios from largest to smallest. This forces the kids to think about the ratios. The kids should always understand why sec >= cos, why csc >= sin, and why cot and tan are not as predictable.

So no, it’s not just some random word salad.

They know that sin is used when they have a hypotenuse and need an “opposite”, or vice versa. The choice of division or multiplication is determined by the SOH triangle.

https://ma610.files.wordpress.com/2013/10/trig-equations.gif

People in general have extremely poor understanding of ratios. They struggle to recognise that 1.2 : 1 is 6 : 5, for example. That sin(57) means the “opposite” to hypotenuse is in the ratio of 0.83867 : 1 is just some random word salad that Maths teachers like to use.

Chester, let me rephrase. Do your kids know what a trigonometric RATIO is. You used the word function, not me. But the function, in right triangle trig, is just returning the ratio. I was asking if your students understood that they were using a ratio of the legs or leg/hypotenuse.

No, of course not.

We teach trigonometry well before we teach functions. It’s not a problem. You can do trig perfectly well without any understanding of functions at all.

I deliberately avoid any mention of ratios when teaching trig. They don’t understand them properly, and it only confuses them.

They still have to memorise that sine starts at 1, don’t they? How do they deal with Tan?

On the whole, the unit circle is the hardest way to teach “understanding” in my experience. For higher level students a trig graph from 0 to 2pi is more effective.

The SOH CAH TOA method of trig triangles pretty much removes any thinking about whether to divide or multiply. It really is foolproof.

But do they know what a trig function is?

For example, I build from special right triangles, where they derive the ratio of the sides, and know that the ratio of short leg to hypotenuse is 1 to 2, etc. Then when I get to right triangle trig, I show them that the ratios are unique to the angle.

That’s not a problem. How would having any more understanding of trig functions aid them in any way, given the sort of problems they have to solve?

I find the students will tend to mix things up terribly with the kind of understanding you are throwing out there, and multiply or divide more or less at random as memory fades.

There’s also later stuff that’s easier or harder to learn based on basic understanding. The fact sine starts at 0 and cosine starts at 1 is tied in with the unit circle meaning in a way that makes it instantly comprehensible to a student with understanding and just another thing to memorize and mix up to a student who is strictly procedural.

Same goes for later identities like sin^2(theta) + cos^2(theta) = 1.

As a pre-calculus teacher I see all the time students clinging to some procedure from geometry (which only has time to do a surface understanding of trig; I know because I also teach geometry) and making my life harder, not easier. I have to actually “de-program” their prior algorithms so they’ll get it right second time around.

Also regarding “what kind of problems they’re going to be responsible for solving”: this one on the PARCC has a fair chance of bowling over any student with only a surface knowledge of sine and cosine.