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# Trigonometry question

lowercase baba
Posts: 12914
64
I was helping my daughter with her trig homework the other day. I'm a bit rusty, admittedly, but I don't know what i'm doing wrong on one problem.

we are given that θ = pi / 3.  We were asked to compute sin (θ^2).

That's pretty straightforward, using a calculator, and getting about 0.8897.

but then things goto weird.  I know that pi/3 = 60 degrees.  When i do substitution, i get sin(60^2), which is sin(3600), or zero.

I know i'm doing something wrong - there must be a hidden assumption i'm making, but i just don't see what I'm doing wrong.

anyone?

Master Rancher
Posts: 3701
44
The error is using degrees. . Do it all in radians and you're good, as you've seen.

The problem when you do it in degrees is that while you may understand the units of θ as degrees, what is θ^2?  It's degrees squared.  What's that?  Well, to make sense of it you need to convert it back to degrees, or to radians.

You may wonder, why didn't we have this problem with radians squared?  Well, radians are really dimensionless - it's an arc length divided by a radius.  Units cancel out.

Marshal
Posts: 70696
288
Is she really supposed to evaluate sinθ²? Not sin²θ? Since sin(π ÷ 3) is √3 ÷ 2, it is really easy to work out its square: ¾.
Neither degrees nor radians have dimensions; does each even have a square at all?

Bartender
Posts: 1201
22

Campbell Ritchie wrote:Is she really supposed to evaluate sinθ²? Not sin²θ? Since sin(π ÷ 3) is √3 ÷ 2, it is really easy to work out its square: ¾.
Neither degrees nor radians have dimensions; does each even have a square at all?

What's the operator precedence?  Does sinθ² equal (sinθ)² or sin(θ²)?  As Campbell implied, sin²θ is shorthand for (sinθ)².

Campbell Ritchie
Marshal
Posts: 70696
288

Ryan McGuire wrote:. . . What's the operator precedence?  Does sinθ² equal (sinθ)² or sin(θ²)? . . .

sin(θ²)

The square operator (like other powers) associates to the right.

Mike Simmons
Master Rancher
Posts: 3701
44

frosenberger wrote: We were asked to compute sin (θ^2).

This seems unambiguous as far as precedence.  However, it's also worth noting that I don't think I've ever seen such a formula actually used for anything, e.g. for physics.  It's far, far more common to see formulas using (sin (θ))^2 - that comes up all over the place.  But sin (θ^2) - not really.  If it's just a made-up example the instructor was using, fine... but it may reflect a mis-transcription of the actual problem.

Campbell Ritchie
Marshal
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Mike Simmons wrote:. . . more common to see formulas using (sin (θ))^2 . . .

I was taught that as sin²θ.

Bartender
Posts: 4109
156
I really don't understand the confusion. If theta is given, then we have no trouble understanding 2 theta, 3 theta, pi theta,... But when we have theta * theta, then it gets confusing?

We have: pi / 3 rad = 60 deg, 2 pi / 3 rad = 2 * 60 deg, ..., pi / 3 * pi / 3 rad = pi / 3 * 60 deg., and not (Freds mistake) 60 * 60 degrees.

Marshal
Posts: 25964
70

Piet Souris wrote:I really don't understand the confusion. If theta is given, then we have no trouble understanding 2 theta, 3 theta, pi theta,... But when we have theta * theta, then it gets confusing?

Yeah, for me it's confusing because there aren't any places in trigonometry where multiplying two angles makes any sense. Okay, if you didn't know any trigonometry then you might just go ahead and try to multiply two angles, but then you fall into the units trap which Mike explained. Sorry, but I'm prejudiced against homework which makes you do pointless things in the name of learning some field of math. If what Fred asked about is really that...

Piet Souris
Bartender
Posts: 4109
156
Well, in highschool we were sometimes confronted with functions like: f(x) = sin(x^2), with the usual questions of calculating intersections with the axes, determinimng limits (if any), doing some differentiation, et cetera. There was no confusion about 'multiplying angles', because all you did was multiplying real numbers.

Campbell Ritchie
Marshal
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I agree with MS about mis‑transcription. I also seem to agree with Paul on the same point.

fred rosenberger
lowercase baba
Posts: 12914
64

Campbell Ritchie wrote:Is she really supposed to evaluate sinθ²?

yes, it truly was sin(θ²).  There were 5-6 problems, and a different one was to evaluate sin²θ

Campbell Ritchie wrote:Neither degrees nor radians have dimensions; does each even have a square at all?

does the number 2 have a dimension?  I think not. but i know it has a square.

and my research shows that the square of a radian is a Steradian, or square radian. It is used in 3-d geometry, referring to a solid angle of a sphere.  It's sort of like the area where a cone starting at the center of a sphere intersects the surface of said sphere.

Campbell Ritchie
Marshal
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I have come across steradians before. The question is obviously to show the difference between the two placements of the square operator x².

Mike Simmons
Master Rancher
Posts: 3701
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Good point about steradians - there are real applications for θ².  Taylor series of trig functions would be another, e.g. cos(θ) = 1 - θ^2 / 2 + θ^4 / 24  - θ^6 / 720 ...  (using radians of course).

But, I can't think of any real-world reason to ever calculate the sine of θ².

It seems like the point of the exercise was to give various examples of functions composed of two or more functions, f(g(x)), which is useful to know about, even if the specific example used doesn't make a lot of sense.

Mike Simmons
Master Rancher
Posts: 3701
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Campbell Ritchie wrote:

Mike Simmons wrote:. . . more common to see formulas using (sin (θ))^2 . . .

I was taught that as sin²θ.

Me too - but for purposes here I wanted to show explicit parentheses to avoid any misunderstanding. Also I was too lazy to look up more specialized character codes.

fred rosenberger
lowercase baba
Posts: 12914
64
i'm probably going to ask the math teacher, too.  I'm sure they hate parents like me.

It probably is a 100% contrived example meant to show the difference between (sin theta)^2 and sin (theta^2)...but it bugs me when I can't see why stuff doesn't work.

Campbell Ritchie
Marshal
Posts: 70696
288

Mike Simmons wrote:. . . I was too lazy to look up more specialized character codes.

Unfortunately, I know the code for ² off by heart: &#xb2; See this old post.

I look forward to hearing what the teacher tells Fred.

Saloon Keeper
Posts: 22671
153

fred rosenberger wrote:I know that pi/3 = 60 degrees.

Then you know wrong. Pi is 3.141592... and is a dimensionless quantity. divide that by 3 (which is also dimensionless) and you get a value slightly larger than 1. Which is a dimensionless value as well.

So your first shot at basic arithmetic bombed.

THEN you have to figure out how to take a non-dimensional value and map it onto a units system - either radians or degrees. In higher math, it's usually radians, but it's better to be certain.

Paul Clapham
Marshal
Posts: 25964
70

Tim Holloway wrote:

fred rosenberger wrote:I know that pi/3 = 60 degrees.

Then you know wrong. Pi is 3.141592... and is a dimensionless quantity. divide that by 3 (which is also dimensionless) and you get a value slightly larger than 1. Which is a dimensionless value as well.

In this context, i.e. angles, Pi represents a number of radians. So yes, Pi/3 is a bit larger than 1 radian. And that number of radians is equal to 60 degrees.

Saloon Keeper
Posts: 12431
269
The whole deal with steradians doesn't make much sense.

Yes, you can square an angle theta in radians to get a surface area measured in steradians (which, like radians is dimensionless, because it measures a fraction of the total area, not an absolute amount), but you would never plug it into the sine function.

Ryan McGuire
Bartender
Posts: 1201
22

fred rosenberger wrote:i'm probably going to ask the math teacher, too.  I'm sure they hate parents like me.

It probably is a 100% contrived example meant to show the difference between (sin theta)^2 and sin (theta^2)...but it bugs me when I can't see why stuff doesn't work.

Fred, any update?

fred rosenberger
lowercase baba
Posts: 12914
64

Ryan McGuire wrote:Fred, any update?

no..I'm sort of torn between wanting to know, and not wanting to embarrass my daughter.  To be honest, I'm satisfied with the "contrived example, and you shouldn't plug steradians into the sine function" answer.

If/when we have parent teacher conferences, i may ask about it, but honestly, i doubt the teacher will know either.  I used to be a secondary school math teacher, and i don't remember ever being taught anything like this...