# Explaining the Why of Trig

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### #1 MamaSprout

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Posted 10 January 2018 - 11:52 AM

We are working on the Trig Chapter of Holt Geometry. It's pretty plug and chug. Dd is fighting me because she wants to know why sin, cos, and tan work. She's seen books of trig tables and wants to know where those numbers come from. Since I have have no idea, I told her to go re-read the Fiendish Angletron and plug and chug for now, and I would find out. Math is easier for her when she gets the whys.

ETA- Can anyone point me to a resource that explains this? Thanks!

Edited by MamaSprout, 10 January 2018 - 11:53 AM.

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Posted 10 January 2018 - 12:16 PM

This Britannica article on the history of trigonometry might help https://www.britanni...ic/trigonometry
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### #3 gstharr

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Posted 10 January 2018 - 12:58 PM

We are working on the Trig Chapter of Holt Geometry. It's pretty plug and chug. Dd is fighting me because she wants to know why sin, cos, and tan work. She's seen books of trig tables and wants to know where those numbers come from. Since I have have no idea, I told her to go re-read the Fiendish Angletron and plug and chug for now, and I would find out. Math is easier for her when she gets the whys.

ETA- Can anyone point me to a resource that explains this? Thanks!

The 12 y.o had Fiendish Angletron by his side when he was working the trig part of alg 2.  It was very helpful. The whole series is a blast.

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### #4 JanetC

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Posted 10 January 2018 - 01:46 PM

Trig tables work based on similar triangles.

If you have a right triangle plus one more given angle theta, it is similar to any other right triangle with that same angle theta.

Let call our two triangles T1 and T2.

Let's call the ratio between them r, such that the length of a side of T2 equals r times the corresponding side of T1.

So, SOH of T1 is

O1/H1

and SOH of T2 is

O2/H2 which is the same as rO1/rH1, and the r's (ratio of similarity) cancel out.

You get the same value for sine theta regardless of how long the actually sides are because the triangles are similar.

That's why you have a table with the trig function values -- the ratios are always the same given the theta.

Edited by JanetC, 10 January 2018 - 01:46 PM.

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### #5 CAJinBE

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Posted 10 January 2018 - 02:07 PM

I think printing out a unit circle with all the common values on it to be very useful.

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### #6 Kathy in Richmond

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Posted 10 January 2018 - 02:13 PM

The Project Mathematics youtube channel has 3 videos explaining trigonometry conceptually. They're oldies, but definitely goodies!

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### #7 daijobu

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Posted 11 January 2018 - 10:00 PM

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### #8 creekland

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Posted 12 January 2018 - 05:48 AM

We are working on the Trig Chapter of Holt Geometry. It's pretty plug and chug. Dd is fighting me because she wants to know why sin, cos, and tan work. She's seen books of trig tables and wants to know where those numbers come from. Since I have have no idea, I told her to go re-read the Fiendish Angletron and plug and chug for now, and I would find out. Math is easier for her when she gets the whys.

ETA- Can anyone point me to a resource that explains this? Thanks!

FWIW, I LOVE teaching students like this - those who really want to know and understand the material.

As to the bolded and underlined... this is true for everyone.  Math is easier when it's understood.  It's no longer memorizing things and hoping your memory is correct.  It's knowing what to do because one understands why it's done.

As for resources, I agree with others at looking at videos or printed versions talking about (and showing) the ratios since that's all trig is.

My Civil Engineering hubby uses Trig more than any other math concept (not counting basic adding, subtracting, multiplying, and dividing of course).

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### #9 MamaSprout

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Posted 12 January 2018 - 07:59 AM

My Civil Engineering hubby uses Trig more than any other math concept (not counting basic adding, subtracting, multiplying, and dividing of course).

I'll tell her this. Right now, she says she wants to be a structural/ civil engineer.

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### #10 MamaSprout

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Posted 12 January 2018 - 08:08 AM

I do!

I struggle with the best way to teach her math. I feel like she would benefit from a more experienced math teacher (than me), but to her, math book open on the floor with a white board is her math happy place, and I can't replicate that with an online class. I know she'd be farther along in math if I had had a better math education. So I just try to stay ahead of her and have 2-3 resources, with videos, available at every level.

Sometimes I'm the only one watching the videos, but I usually can answer her questions when they come up this way. This one kind of threw me because we aren't really to the trig chapter yet... she wanted to learn it to better understand what she's doing in physics, which is teaching just enough trig to do the physics. Math around here is a little like the Give a Mouse a Cookie books.

As long as no one cries, it's all good.

Thanks for all of the resources! ETA- I'm teaching her the Unit Circle this morning.

Edited by MamaSprout, 12 January 2018 - 08:10 AM.

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### #11 creekland

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Posted 12 January 2018 - 08:25 AM

I do!

I struggle with the best way to teach her math. I feel like she would benefit from a more experienced math teacher (than me), but to her, math book open on the floor with a white board is her math happy place, and I can't replicate that with an online class. I know she'd be farther along in math if I had had a better math education. So I just try to stay ahead of her and have 2-3 resources, with videos, available at every level.

Sometimes I'm the only one watching the videos, but I usually can answer her questions when they come up this way. This one kind of threw me because we aren't really to the trig chapter yet... she wanted to learn it to better understand what she's doing in physics, which is teaching just enough trig to do the physics. Math around here is a little like the Give a Mouse a Cookie books.

As long as no one cries, it's all good.

Thanks for all of the resources! ETA- I'm teaching her the Unit Circle this morning.

I suspect with her love and drive plus your willingness to find resources she'll do just fine.  Unfortunately, not all math teachers are as capable at actually teaching math as we'd like to think.  Many teach memorization.  One thing I often do when helping students is start off teaching the "why."  Sadly, the vast majority (needing help) are never taught this.  I can run into seniors who don't know what a square root is aside from a button on the calculator.  The fact that it IS something (the length of a side of a square with the area they're taking the root of) is so simple, but unknown.  Same thing with division by zero (why we can't do it), trig functions, and oodles more.

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### #12 kiana

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Posted 12 January 2018 - 08:46 AM

I suspect with her love and drive plus your willingness to find resources she'll do just fine.  Unfortunately, not all math teachers are as capable at actually teaching math as we'd like to think.  Many teach memorization.  One thing I often do when helping students is start off teaching the "why."  Sadly, the vast majority (needing help) are never taught this.  I can run into seniors who don't know what a square root is aside from a button on the calculator.  The fact that it IS something (the length of a side of a square with the area they're taking the root of) is so simple, but unknown.  Same thing with division by zero (why we can't do it), trig functions, and oodles more.

Yes.

I was just writing a teaching statement for job applications and was discussing strategies I use for moving students towards a conceptual basis when they have been taught and are only comfortable with an algorithmic basis.

Dividing by zero is a perfect example. They learn that "Anything divided by zero is zero" and what sticks in their head is "0 is BAD in division problems". So when they see "0/5", they immediately go to "undefined!"

Another place where it's very obvious is in absolute value. They've learned, on an algorithmic level, that |x| = c has the solutions +/- c. So when we ask for solutions of |x| = -4, instead of saying "But the absolute value can't be negative! There can't be solutions!" they answer -/+ 4.

tl;dr OP's dd is going to be a pleasure to teach in college

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### #13 SparklyUnicorn

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Posted 12 January 2018 - 09:05 AM

I think printing out a unit circle with all the common values on it to be very useful.

I personally don't find this useful for helping me understand the why behind trig.

I can be rather dense though.  LOL

### #14 JanetC

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Posted 12 January 2018 - 10:59 AM

So sine and cosine both have the hypotenuse in the denominator. On the unit circle, the hypotenuse is one, which makes the division step trivial. That's the reason for using it.

### #15 CAJinBE

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Posted 12 January 2018 - 02:37 PM

I personally don't find this useful for helping me understand the why behind trig.

I can be rather dense though.  LOL

It helps to see the relationships between angles and the values for cosine, sine, and tangent very visually. It also helps to see the relationship between radians and degrees. It also helps to see why there can be more than one angle with the same cosine value for example, especially on those problems where you add multiples of 2pi. When I get confused, I would rather look at a unit circle than a trig table or a value on my calculator.

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