Why can't we apply trig ratios to any angle of a right triangle?

[musicgold]

Why do trigonometric ratios have to be related to the angle between the base and hypotenuse of a right angle triangle?

I am trying to understand why I can't use these ratios to any angle of a right angle triangle. I try to do that in the attached document. It seems to work for all ratios except cos90 and tan90.

I know the origin of the trigonometric ratios is in the unit circle and the length of the cord, but why can't we extend this principal to any angle of a right angle triangle?

Thanks.

 

[SteamKing]

The cosine of 90 degrees is zero and the tangent of 90 degrees is infinite. What's wrong with that?

 

[AlephZero]

It is true that sin B = 1, because B = 90 degrees.

But I don't understand why you said "sin B = k/k".

 

[Chestermiller]

The answer is that the trigonometric ratios are defined only for the non-right angles of a right triangle.

 

[musicgold]

The cosine of 90 degrees is zero and the tangent of 90 degrees is infinite. What's wrong with that?

Cos B = adjacent side / hypotenuse

To find Cos B, I am not sure which adjacent side to use - it can be h or b.

Either way, the ratios ( h/k or b/k) are not going to be zero. So I was not sure what to do.

It is true that sin B = 1, because B = 90 degrees.

But I don't understand why you said "sin B = k/k".

Sin B = opposite side / hypotenuse

In this case both of them are k.

The answer is that the trigonometric ratios are defined only for the non-right angles of a right triangle.

Ok. I didn't know that. Where can I read more on this?

Thanks.

 

[Ray Vickson]

Cos B = adjacent side / hypotenuse

To find Cos B, I am not sure which adjacent side to use - it can be h or b.

Either way, the ratios ( h/k or b/k) are not going to be zero. So I was not sure what to do.




Sin B = opposite side / hypotenuse

In this case both of them are k.



Ok. I didn't know that. Where can I read more on this?

Thanks.

Adjacent side = side adjacent to the angle = side 'coming out of the angle'.
Opposite side = side opposite the angle = side the angle does not 'touch'.

Isn't this material in your textbook or course notes? If not, see http://www.purplemath.com/modules/basirati.htm . Look at the diagram.

 

[Chestermiller]

Cos B = adjacent side / hypotenuse

To find Cos B, I am not sure which adjacent side to use - it can be h or b.

Either way, the ratios ( h/k or b/k) are not going to be zero. So I was not sure what to do.




Sin B = opposite side / hypotenuse

In this case both of them are k.



Ok. I didn't know that. Where can I read more on this?

Thanks.

It should be in any plane geometry textbook. Maybe they never thought to mention that these definitions don't apply to the right angle of the triangle.

Chet

 

[HallsofIvy]

The answer is that the trigonometric ratios are defined only for the non-right angles of a right triangle.

That is true only for the "trigonometric ratios" as defined in a trig course, in terms of a right triangle. The trig functions, sin(x) and cos(x), are defined for all x. The tangent and cosec functions are defined for all x such that [itex]sin(x)\ne 0[/itex]. The cotangent and secant functions are defined for all x such that [itex]cos(x)\ne 0[/itex].

They can be defined completely independently of "triangles" using power series for example.

 

[Chestermiller]

That is true only for the "trigonometric ratios" as defined in a trig course, in terms of a right triangle. The trig functions, sin(x) and cos(x), are defined for all x. The tangent and cosec functions are defined for all x such that [itex]sin(x)\ne 0[/itex]. The cotangent and secant functions are defined for all x such that [itex]cos(x)\ne 0[/itex].

They can be defined completely independently of "triangles" using power series for example.

This, of course, is correct, but it doesn't seem to address the question in the context that OP was asking it.

Chet

 

[gopher_p]

... tangent of 90 degrees is infinite.

This is not true. Tangent of 90 degrees is undefined.

 

[SteamKing]

This is not true. Tangent of 90 degrees is undefined.

But it's still infinite.

 

[gopher_p]

But it's still infinite.

Tangent of 90 degrees is not defined. It is not a thing. It is mathematically meaningless; like "length of a circle" or "square root of cow". It does not have properties. In particular it does not have the property of being infinite. I don't know who told you that it did, but they were wrong.

 

[SteamKing]

When I look at a reference like Abramowitz and Stegun for the tangent of 90 degrees, I don't see 'undefined' or the 'square root of a cow' listed; I see a lazy 8, which to me means infinity. Now, I agree that mathematical operations involving infinity are undefined, but if the limit of a series tends to infinity, we don't say that the series is 'undefined', we say it 'diverges'.

BTW, I take the 'length of a circle' to means its circumference.

 

[jackarms]

Here's a good way to think about it: when you apply the ratios to right angles, essentially you're going to have a right triangle where the angle you have the ratio on is also a right angle. Now obviously this shape wouldn't actually be a triangle, but using the ratios at 90 degrees is more like seeing what the ratios approach when one of the two other angles approaches 90.

 

[gopher_p]

When I look at a reference like Abramowitz and Stegun for the tangent of 90 degrees, I don't see 'undefined' or the 'square root of a cow' listed; I see a lazy 8, which to me means infinity.

This source also says ##\ln 0=-\infty##, ##e^\infty=\infty##, and ##e^{-\infty}=0##. It also defines the "circular functions" in terms of complex exponentials. It uses the lemniscate repeatedly, and yet I can't for the life of me find where they actually say what they mean when they use it. I'm going to go out on a limb and say that it's not entirely appropriate as a reference for precalculus mathematics.

Under just about any definition of tangent appropriate for non math majors, "tangent of ninety degrees" is virtually meaningless in that it does not have the same meaning as "tangent of x" for any other (appropriate) value of x.

Now, I agree that mathematical operations involving infinity are undefined, but if the limit of a series tends to infinity, we don't say that the series is 'undefined', we say it 'diverges'.

Right. We aren't talking about limits though. We're talking about right triangle trigonometry. If you want to talk about limits of real-valued functions, then you should say that. But since you brought up series, I'll remind you of the recent internet kerfuffle regarding ##\sum n=-1/12##. While there is a context in which that is true, it is not part of the "standard" interpretation of infinite sums. Furthermore, it would be misleading and (if you actually know what you're talking about) willfully dishonest to tell someone that the result of adding all of the positive integers together gives you -1/12.

What I am trying to convey to you is that it is dishonest and misleading to tell a precalculus student that tangent of 90 degrees is infinite.

BTW, I take the 'length of a circle' to means its circumference.

Oops. I meant to say sphere.