kitchen table math, the sequel: help desk - sine and arcs and circular functions

## Thursday, April 21, 2011

### help desk - sine and arcs and circular functions

From Dolciani's Algebra and Trigonometry, Chapter 13-2 Circular Functions, p. 555:
sin s = y
cos s = x
tan s = sin s/cos s if cos s ≠ 0
cot s = cos s/sin s if sin s ≠ 0
sec s =1/cos s if cos s ≠ 0
csc s = 1/sin s if sin s ≠ 0
s is the length of an arc of a circle.

This may be asking too much, but I need help.

I have never seen sine, cosine, tangent, etc. applied to an arc. I've only learned sine and cosine in relation to angles in a right triangle.

I've looked back through chapter 12, but I don't see a section that explains this. I'm sure it's there, but it's not obviously there, and I'm in a hurry, sad to say.

Is there a short way anyone can explain to me how we get from sine, cosine, angles, and SOHCAHTOA to sine, cosine, and arcs?

Is there a website that has a succinct and lucid explanation?

And is there a book you like for self-teaching trigonometry and algebra 2? (Do we know what book(s) homeschoolers use?)

I need a royal road to circular functions.

Allison said...

I think the problem is they just did. :)

Let's back up.

"I have never seen sine, cosine, tangent, etc. applied to an arc. I've only learned sine and cosine in relation to angles in a right triangle."

sine, cosine, and tangent are applied to ANGLES. You can take the sine of an ANGLE.

Sine is a function whose domain is the range of values an angle can take, and whose domain is bounded between 1 and -1.

Does that make sense? The issue isn't the triangle per se. The issue is the ANGLE, which was an angle inside a triangle, yes, but the relevant issue is the ANGLE.

Btw, do you own a geometry book? Have you looked in there for sine and cosine as functions?

Allison said...

(cont)

Something helpful to look at would be the unit circle. Heard of it? Seen it?

http://en.wikipedia.org/wiki/Unit_circle

Start there. Tell us what you recognize.

ChemProf said...

Allison is right -- you need to start with the unit circle, or circle of radius 1. For a unit circle, the length of an arc is the same as the angle in radians (since the circumference is 2 pi r or just 2 pi).

Check the wikipedia and check if Dolciani has a chapter on the unit circle, then see if the rest makes sense again.

Bostonian said...

I agree with the previous posters that that unit circle is very important in understanding the sine and cosine functions. Knowing that cos(theta) and sin(theta) are the x and y values on the unit circle, you can deduce

what sin(0) and cos(0) are

what the periodicity of sine and cosine are

at what arguments sine and cosine have maxima and minima

which function is odd and which is even

where sin(theta) = cos(theta)

why (sin(theta))^2 + (cos(theta))^2 = 1

Michael Weiss said...

On any circle, the length of an arc (measured in units of the radius) is the same as the measure of the corresponding central angle (measured in radians). In fact that's how radian measure is defined: take an angle, draw a circle centered at the vertex of the angle, measure the length of the arc and divide it by the radius. That's why a "90 degree" angle is a "pi/2 radians"
angle; because the length of a quarter-circle will be exactly pi/2 times the radius.

It's customary to think of "unit circles", in which the radius of the circle is 1. [One what? Why, one radius, of course. :) ]. In such a case the arc length is the angle measure: the two are really identical. In that context, the passage from the textbook makes sense.

Anonymous said...

Because you are working with a circle of unit radius

(a) the circumference of the circle is 2 x pi units
(b) the total angle around the circle is 360 degrees, or 2 x pi radians

so as a previous commenter said, 'sine s' where s is the length of the arc is the same thing as the sine of the angle subtended at the centre of the circle by the arc

And if you draw a little sketch of x and y axes, with a circle of unit radius centred at the origin, and an arc that runs in the top right quadrant from the x axis towards the y axis, and draw the perpendiculars from the end of the arc to the x and y axes, you will see that your 'opposite' length of to plug into your formula is the y co-ordinate of the end of your arc, and the 'adjacent' length is the x co-ordinate of the same point.

I am not sure that I have made things any clearer, but I have tried . . . .

Catherine Johnson said...

thanks!

tutor has just been here and gone -- I was missing the unit circle --

The Regents prep page is terrific for someone in my position --

Here it is: arc length and radian measure

Catherine Johnson said...

if you draw a little sketch of x and y axes, with a circle of unit radius centred at the origin, and an arc that runs in the top right quadrant from the x axis towards

Thank you!

I did that!

Catherine Johnson said...

Is it 'wrong' to learn radian measurement before you know anything about unit circles?

(This has happened entirely because we have a math fiasco and major tutoring needs going on around here. Normally I wouldn't jump into Chapter 13 before Chapters 1-12....)

Allison said...

Basically, yes, it's wrong because you can't define a radian without a circle.
Consider a circle with radius r. Take the radius and wrap it around the edge of the circle. The angle that forms along that arc is a radian.

Anonymous said...

Happy to help and glad the sketch idea was useful - I should have said also that because the circle has unit radius, the hypotenuse is always 1.

Anonymous said...

I think this explanation is great: http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/

Michael Weiss said...

Is it 'wrong' to learn radian measurement before you know anything about unit circles?

A unit circle is just a circle whose radius is 1. And, as I said somewhat obscurely above, the question "1 what?" is usually not specified -- we do not bother with the units on the unit circle, so to speak. So as a practical matter any circle is a unit circle, where the radius is "1 radius long".

So there really isn't anything you need to know about unit circles, other than that they are circles, and that the radius is 1.

AS's explanation is good. I would phrase it slightly differently: Choose any point on the unit circle in the 1st quadrant (positive x, positive y). Draw in the radius that goes from that point to the origin. Now drop a perpendicular down to the x-axis to form a right triangle.

In an Algebra 1 class, you would now start talking about "rise", "run", and "slope."

Instead, we are going to focus on the angle with one side on the posiive x-axis, and the other side pointing directly though the point you chose on the circle.

Now, the "rise" is just the y-coordinate of your chosen point, which is also the "opposite" side of the angle.

The "run" is just the x-coordinate of your chosen point, which is also the "adjacent" side of the angle.

The hypotenuse is just the radius of the circle, which is 1 because this is a unit circle.

So cos(theta) is the x-coordinate of your point, sin(theta) is the y-coordinate of your point, and tan(theta) is y/x, also known as rise over run, also known as the slope of the radius.

Catherine Johnson said...

Oh - thanks for the Better Explained link -- I'd forgotten all about that site!

Catherine Johnson said...

A unit circle is just a circle whose radius is 1. And, as I said somewhat obscurely above, the question "1 what?" is usually not specified -- we do not bother with the units on the unit circle, so to speak. So as a practical matter any circle is a unit circle, where the radius is "1 radius long".

Got it!

(at least, I think I've got it -- !)

Anonymous said...

I found the Art of Problem Solving "Precalculus" book to have an excellent explanation of trignometry. They think about it the way I do, and it was a good fit for my son.

I don't know whether it would work for you Catherine, but it might be worth a try.

Catherine Johnson said...

oh no!

I wish you hadn't told me that....

I've just decided that the Art of Problem Solving books are getting on my nerves --- !

Now I'm going to have to buy another one!

Seriously, though, thanks very much for the tip. (It's just funny that I'm getting an AofPS recommendation at the very moment I've grown exasperated with the "Problem Solving" organization of the books!)