When is it talking about degrees or radians?

[questionposter]

So I've noticed that based on problems I've worked on, whenever the degree is known, for some reason I should automatically assume there's a unit circle involved, and when the degree isn't known and the angle is just left as theta, then the problem is for some reason automatically talking about degrees instead, but I have no idea why. So what if the angle is known? I don't see a unit circle magically appearing when I'm measuring the angle of a window or whatever.

 

It's something like this, it's not identity, it's just trigonometric functions.

 

 

Givens:

 

sin30°=1/2 tan30°=√(3)/3

 

What I need to find:

 

a. csc30° which on a unit circle is 2

b. cot60° 30-60-90 triangle on a unit circle, by answering "c." first, I came up with (1/2)/√(3/4) which I believe simplifies to (4√(3/4))/3

c. cos30° which because of the Pythagorean theorem is 1²-(1/2)² = .75 or 3/4, so cos=√(3/4)

d. cot30° which is √(3/4)/(1/2) which I believe simplifies to 2√(2/4)

 

I mean I know those coordinates correspond to a point on a unit circle, but why do I automatically have to talk about radians just because I know the angle? If the sin is 1/2, couldn't the side lengths be any ratio of size that forms 1/2, like the side lengths are 2 and 4, or 8 and 16? Why do the givens automatically become the side lengths when the actual side lengths could be any numbers that have a ratio of 1/2? Couldn't I have 1/4 and 1/2 as side lengths too?

Edited November 6, 2011 by questionposter

[Cap'n Refsmmat]

I'm not sure I understand your question. The unit circle is merely a helpful way to remember the values of trig functions at different angles; the side lengths can indeed be any sizes that form the same ratio.

 

Could you clarify your question?

[DrRocket]

The unit circle is there and has the same significance whether you are measuring angles in degrees or radians.

 

The utility of radians is that they are mathematically "natural" because the measure of an angle in radians is precisely the arc length of the subtended arc on the unit circle. It is also natural to think of the unit circle in the complex plane, which is a group under the operation of multiplication. The utility of degrees is that the angle is easily represented to an accurate approximation in terms of an integer number of degrees, minutes and seconds (you don't have to deal directly with the transcendental number pi).

 

Mathematicians tend to use radians. Engineers and physicists use degrees or radians as convenient to a specific application.

Edited November 6, 2011 by DrRocket

[questionposter]

The unit circle is there and has the same significance whether you are measuring angles in degrees or radians.

 

The utility of radians is that they are mathematically "natural" because the measure of an angle in radians is precisely the arc length of the subtended arc on the unit circle. It is also natural to think of the unit circle in the complex plane, which is a group under the operation of multiplication. The utility of degrees is that the angle is easily represented to an accurate approximation in terms of an integer number of degrees, minutes and seconds (you don't have to deal directly with the transcendental number pi).

 

Mathematicians tend to use radians. Engineers and physicists use degrees or radians as convenient to a specific application.

 

Well, if the given is just a ratio, how does the triangle have only 1 exact answer per side length of a triangle? Because when I look up the answers, they are specific numbers, but that can't be right because 1/2 is the same as 2/4 or 3/6, I could have any number of side lengths.

 

The way it works is, if the given is theta, like tanΘ = 5/1, then the opposite of the triangle is 5 units and the base is 1 unit in length, but couldn't it be 10 and 2 or 1 and (1/5)? Why does it assume that? How could that ever work in the real world?

Edited November 6, 2011 by questionposter

[Cap'n Refsmmat]

What do you mean, one exact answer? Do you mean one exact value for sine, cosine, etc.?

 

The trigonometric functions are ratios of sides, so it doesn't matter how long the sides are, so long as the ratio between them is the same. So [imath]\sin \frac \pi 3[/imath] has the same value, regardless of whether the hypotenuse is 1 unit or 23 units long, since the ratio between the two sides will be the same.

[questionposter]

What do you mean, one exact answer? Do you mean one exact value for sine, cosine, etc.?

 

The trigonometric functions are ratios of sides, so it doesn't matter how long the sides are, so long as the ratio between them is the same. So [imath]\sin \frac \pi 3[/imath] has the same value, regardless of whether the hypotenuse is 1 unit or 23 units long, since the ratio between the two sides will be the same.

 

So if the only information I'm given is sinΘ=1/2 or tanΘ=1/5, those are just ratios, how could I only have one set of side lengths from that? It could be a bunch of different side lengths, so why is it only assuming one possible set of side length? When could that assumption ever come on handy?

 

So if the given is cosΘ=1/3, then when I look up the answers, the side lengths of the triangle are adjacent=1 and hypotenuse=3, but it could be a bunch of different numbers, not just 1 and 3. It could be 2 and 6, or 1 and 1/3, or 1/3 and 2/6 and etc.

 

Oh wait a minute, I think it's asking me to find the trigonometric functions, therefore in order to find the corresponding identities and co-functions, I have to use a specific set of side lengths that have the same ratios as the given, so my goal isn't to evaluate the lengths of the triangle, its just to find the functions and co-functions, which means I need get them by using real side lengths right?

 

Huh, I didn't know it automatically merged double posts.

Edited November 6, 2011 by questionposter

[DrRocket]

Well, if the given is just a ratio, how does the triangle have only 1 exact answer per side length of a triangle? Because when I look up the answers, they are specific numbers, but that can't be right because 1/2 is the same as 2/4 or 3/6, I could have any number of side lengths.

 

The way it works is, if the given is theta, like tanΘ = 5/1, then the opposite of the triangle is 5 units and the base is 1 unit in length, but couldn't it be 10 and 2 or 1 and (1/5)? Why does it assume that? How could that ever work in the real world?

 

That is one reason why the UNIT circle is used for convenience.

[Cap'n Refsmmat]

So if the only information I'm given is sinΘ=1/2 or tanΘ=1/5, those are just ratios, how could I only have one set of side lengths from that? It could be a bunch of different side lengths, so why is it only assuming one possible set of side length? When could that assumption ever come on handy?

The only way you can do this is if you know the length of one side of the triangle. You often know the length of one side and one angle, and want to find the length of the other sides. Trigonometric functions give the ratios you can use to find them.

 

Oh wait a minute, I think it's asking me to find the trigonometric functions, therefore in order to find the corresponding identities and co-functions, I have to use a specific set of side lengths that have the same ratios as the given, so my goal isn't to evaluate the lengths of the triangle, its just to find the functions and co-functions, which means I need get them by using real side lengths right?

I don't understand what you're asking. As far as I can tell, it's asking you to find find the values of csc and cot and such. The values of those trig functions evaluated at a specific angle are ratios between lengths. They are not side lengths.

[questionposter]

I don't understand what you're asking. As far as I can tell, it's asking you to find find the values of csc and cot and such. The values of those trig functions evaluated at a specific angle are ratios between lengths. They are not side lengths.

 

Yeah, I figured it out. My goal isn't to find the side lengths, its to find the trigonometric properties of a triangle that uses the given. So in a 30-60-90 triangle, the only way I could devise a theorem for it is to relate the side lengths, so I need to find a real 30-60-90 triangle in order to figure out how the side lengths relate, and how those side-lengths relate is what I'm trying to find, not the side lengths themselves. So no matter what, the hypotenuse of a 30-60-90 will be 2x, the base will be x, and the other leg will be √(3)*x, but I need to use real side lengths in a real 30-60-90 triangle in order to figure out that pattern in the first place, only in the case of the problems, I'm only given one property of a triangle and I have to figure out the rest based on that one property, which includes using the side lengths that the given property generates.

 

Thanks for trying to help though.

 

So basically, I CAN pick any number of side lengths as long as they all relate in the way they are suppose to. The answer book was just using the simplest side lengths to relate.

Edited November 6, 2011 by questionposter

[Cap'n Refsmmat]

So basically, I CAN pick any number of side lengths as long as they all relate in the way they are suppose to. The answer book was just using the simplest side lengths to relate.

Right. You can only determine uniquely a side length if you already know some other property of the triangle, like the length of another side.