What Is a Double Angle and Why Is It Important in Mathematics and Physics?

[DecayProduct]

I'm having trouble visualizing double angles. What is the physical nature of double angles? I have books which talk about how to do math on double angles. I have seen websites about how to do math on double angles. But I have yet to find out WHAT a double angle really means. Where do they occur in the physical world, or in the physical equations that govern the world?

What the heck is a double angle!?

 

[Defennder]

You mean the trigo double angle formulae or something else?

 

[DecayProduct]

You mean the trigo double angle formulae or something else?

Not so much the formulae, but the THING the formulae are describing. But yes, I'm teaching myself math, and now I'm into trig. And my trig has just entered into double angles, but I can't seem to get a picture in my head what a "double angle" looks like.

 

[HallsofIvy]

? As far as I know a "double angle" just means "twice an angle". In other words, the "double angle formulas", sin(2x)= 2 sin(x)cos(x) and cos(2x)= cos²(x)- sin²(x), just tell you how, if you already know sin(x) and cos(x), to calculate sin(2x) and cos(2x). For example, if you know that sin(30)= 1/2 and cos(30)= sqrt(3)/2, then you can calculate that sin(60)= sin(2*30)= 2*sin(30)cos(30)= 2(1/2)(sqrt{3}/2)= sqrt(3)/2 and that cos(60)= cos²(30)- sin²(30)= (sqrt(3)/2)²- (1/2)²= 3/4- 1/4= 1/2.

If you mean something other than just "an angle twice as large" I would like to see a problem or statement from a textbook about it!

 

[DecayProduct]

? As far as I know a "double angle" just means "twice an angle". In other words, the "double angle formulas", sin(2x)= 2 sin(x)cos(x) and cos(2x)= cos²(x)- sin²(x), just tell you how, if you already know sin(x) and cos(x), to calculate sin(2x) and cos(2x). For example, if you know that sin(30)= 1/2 and cos(30)= sqrt(3)/2, then you can calculate that sin(60)= sin(2*30)= 2*sin(30)cos(30)= 2(1/2)(sqrt{3}/2)= sqrt(3)/2 and that cos(60)= cos²(30)- sin²(30)= (sqrt(3)/2)²- (1/2)²= 3/4- 1/4= 1/2.

If you mean something other than just "an angle twice as large" I would like to see a problem or statement from a textbook about it!

The texts I have read never just say so simply that, say, "sin 2x" equals "the sine of an angle twice as large as x". If that is the case, then a question that enters my mind is, why would anyone need the blasted formulae for double angles? What I mean is, if I know the angle is x, and I wanted to know sin 2x, why wouldn't I just calculate the sine of that angle? Instead of saying sin 2x = 2sinxcosx?

 

[roam]

Hi!

I didn't quite understand but there are 3 important double angle formulas; for sin(2A), cos(2A) and tan(2A).

Is sin(2A) = 2 sin(A)? No

An obvious counter example is that sin(60°) which is 0.8660 sin not equal to to 2 × sin(30°), which is 2 × 1/2 =1

In fact, sin(2A) = 2sin(A)cos(A). I'll show you the proof;

sin(2A)
consider the compund angle formula: sin(A+B) = sin(A)cos(B)+cos(A)sin(B)
Substitute A for B:
sin(A+A)= sin(A)cos(A)+cos(A)sin(A)
sin(2A) = 2sin(A)cos(A)

End of Proof

 

[snipez90]

The texts I have read never just say so simply that, say, "sin 2x" equals "the sine of an angle twice as large as x". If that is the case, then a question that enters my mind is, why would anyone need the blasted formulae for double angles? What I mean is, if I know the angle is x, and I wanted to know sin 2x, why wouldn't I just calculate the sine of that angle? Instead of saying sin 2x = 2sinxcosx?

Because not everything is centered around simply computing a value. Say you had to solve the equation sqrt(1-sin(2x)) + sqrt(1+sin(2x)) = cos(x). Are you going substitute random x values until you reach equality? Using sin(2x) = 2sinxcosx helps us simplify this expression.

If you need to show a trig identity, the double angle formula also helps. There isn't much to visualize since you can prove this result easily from the sin addition formula. I know there are at least 2 geometric proofs for the sin addition formula which is the closest thing to "visualizing" the theorem.

The above example is a bit contrived. Yesterday I was solving an integral and I got to the point where I had to find the indefinite integral of sin(2t)cot(t)dt. Knowing the identity immediately came down to finding the integral of [cos(t)]^2 which is well known.

 

[dharries]

I had the same problem of trying to visualize what double/half angles do. After i read this article,

http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/DoubleAngle-and-HalfAngle-Identities.topicArticleId-11658,articleId-11612.html

it made complete sense to me! read first what it says than study the graphs on it. Good luck!

 

[dynamicsolo]

...why would anyone need the blasted formulae for double angles? What I mean is, if I know the angle is x, and I wanted to know sin 2x, why wouldn't I just calculate the sine of that angle? Instead of saying sin 2x = 2sinxcosx?

Ah, Child of the Age of Electronic Calculation Aids, these formulas were worked out long before there were devices for obtaining trig values at the touch of a button or keyboard... The various identities were extremely useful in saving time working out values of trig functions for new angles from values that were already known for other angles. Before about 1973, unless you had frequent access to a mainframe computer (not likely or terribly convenient in any case), you either got (very rough) trig values from a slide rule, or you used a book of trig tables. Those tables had to be computed by people using the trigonometric identities (and interpolation rules, etc.) and an (electro-)mechanical calculator, or, before the late 19th Century, by hand!. So a lot of these equations that turn up were incredibly useful to get most trig functions people needed.

Enough of the history lecture. Nowadays, while you don't use most of these identities for the same purposes they were developed, some of them are still helpful in certain ways. The double-angle formulas for sine and cosine turn out to be valuable in making a certain substitution in order to integrate the functions sine-squared and cosine-squared, as well as many others, just as one example. There are also a number of physics problems where double-angle relations turn up; for instance, on level ground, the range of a projectile fired with velocity v at angle theta to the horizontal (neglecting air resistance) is

[tex] R = \frac{v^2 \cdot sin(2\theta)}{g}[/tex]