(Seven deadly) sines and cosines, part I

Illustrative picture

Hello outside world! I realize it’s been more dead than alive around here lately. The thing is – it’s summer time and as far as my War on Procrastination goes, I am getting brutally massacred. Yep, that’s it – I’m mostly wasting my time and potential. Don’t do that, it’s not a good thing.

Anyway, until this horrible dark age is over and I (hopefully) gather some inspiration, I will provide you with some purely mathematical material. Tomáš, an old friend and a high school classmate of mine, currently studying Financial Mathematics at Masaryk University has always been kind of a fan of trigonometry. That’s why he has provided me with a list of trigonometric identities he finds useful during his studies, together with their proofs. I have divided this list into several articles, each containing seven formulas and their proofs. Yes, the heading pun was intended :).

We’ll start with the most trivial and simple ones, but I guess it’ll get more interesting later on. Let’s begin.

(1) Pythagorean identity

Statement: Let \theta \in [0,2\pi) . Then

\sin^2 \theta + \cos^2 \theta = 1 .


Proof: We know that the equation x^2 + y^2 = 1 defines a unit circle centered at (0,0) in Cartesian coordinates. By transforming to polar coordinates, we get for every point (x,y) of this circle that x = \cos \theta and y = \sin \theta for some \theta \in [0,2\pi) and thus \sin^2 \theta + \cos^2 \theta = 1 .

Note: By using the periodicity of sine and cosine, the above formula can actually be extended for any real \theta .


(2) Angle sum identities

Statement: Let \alpha, \beta be any real numbers. Then

  • \sin (\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta ,
  • \cos (\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta .

While there are many ways to approach this and some geometric proofs are indeed elegant, I’m not much of a friend of geometry and so I chose a more elegant, though rather complex (pun intended) way.


Proof: We will use the Euler’s formula. We have that e^{i\alpha} = \cos \alpha + i\sin \alpha, \; e^{i\beta} = \cos \beta + i\sin \beta . Therefore

\cos(\alpha+\beta) + i \sin(\alpha + \beta) = e^{i(\alpha+\beta)} = e^{i\alpha}e^{i\beta} =

= (\cos \alpha + i\sin \alpha)(\cos \beta + i \sin \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta +

+ i( \sin \alpha \cos \beta + \cos \alpha \sin \beta)

Since two complex numbers are equal exactly when their respective real and imaginary parts are equal, we get

\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta,

\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta.

Substituting -\beta for \beta and using the fact that sine is an odd function (which means that \sin (-\beta) = -\sin \beta ) and cosine an even one (thus \cos (-\beta) = \cos \beta ) proves the other two identities.


(3) Sine of double angle

Statement: Let \theta be any real number. Then

\sin(2\theta) = 2\sin\theta\cos\theta .


Proof: We use the angle sum formula from (2) for sine to easily get our result:

\sin(2\theta) = \sin(\theta + \theta) = \sin \theta \cos \theta + \cos \theta \sin \theta = 2\sin\theta\cos\theta .


(4) Cosine of double angle

Statement: Let \theta be any real number. Then

\cos(2\theta) = \cos^2 \theta - \sin^2 \theta .


Proof: The same way as before, we use the angle sum formula for cosine from (2):

\cos(2\theta) = \cos(\theta + \theta) = \cos \theta \cos \theta - \sin \theta \sin \theta = \cos^2 \theta - \sin^2 \theta .


(5) Sine of half-angle

Statement: Let \theta be any real number. Then

\left| \sin \frac{\theta}{2} \right| = \sqrt{\frac{1-\cos \theta}{2}} .


Proof: We know from (1) and (4) that for any x

\sin^2 x + \cos^2 x =1

and

\cos^2 x - \sin^2 x = \cos 2x .

Subtracting these equations, we get that

2\sin^2 x = 1-\cos 2x .

and thus

\left| \sin x \right| = \sqrt{\frac{1-\cos 2x}{2}} .

By substituting x = \frac{\theta}{2} we get the result.

Note: I find this identity much more useful in the form \sin^2 x = \frac{1-\cos 2x}{2} . For example, when you’re working with trigonometric integrals, this little tricks reduces the power of a function by one, which often comes very handy.


 (6) Cosine of half-angle

Statement: Let \theta be any real number. Then

\left| \cos \frac{\theta}{2} \right| = \sqrt{\frac{1+\cos \theta}{2}} .


Proof: We proceed the same way as in the previous proof, except this time we add up the two equations instead of subtracting them. Thus, we get

2\cos^2 x = 1+\cos 2x

and

\left| \cos x \right| = \sqrt{\frac{1+\cos x}{2}} .

Again, substituting x=\frac{\theta}{2} proves the desired identity.

Note:  \cos^2 x = \frac{1+\cos x}{2} is also a more useful form of this identity, see the note after (5).


(7) Sine sum-to-product identities

Statement: Let \alpha, \beta be any real numbers. Then

  • \sin \alpha + \sin \beta = 2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} ,
  • \sin \alpha - \sin \beta = 2 \cos \frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2} .

Proof: We will prove both of these identities at once. First of all, note that

  • \alpha = \frac{\alpha+\beta}{2}+\frac{\alpha-\beta}{2}
  • \beta = \frac{\alpha+\beta}{2}-\frac{\alpha-\beta}{2}

Knowing this, we can use the angle sum identity from (2) to arrive at the result:

\sin \alpha \pm \sin \beta = \sin \left( \frac{\alpha+\beta}{2} + \frac{\alpha-\beta}{2}\right) \pm \sin \left(\frac{\alpha+\beta}{2} - \frac{\alpha-\beta}{2} \right) =

= \sin \left( \frac{\alpha+\beta}{2} \right) \cos \left( \frac{\alpha-\beta}{2} \right) + \cos \left( \frac{\alpha+\beta}{2} \right) \sin \left( \frac{\alpha-\beta}{2} \right) \pm

= \left( \sin \left( \frac{\alpha+\beta}{2} \right) \cos \left( \frac{\alpha-\beta}{2} \right) - \cos \left( \frac{\alpha+\beta}{2} \right) \sin \left( \frac{\alpha-\beta}{2} \right) \right) .

We get the desired formulas by substituting respective + or – sign into the last equation.


So, that’s it for today. I am aware that these are mostly very simple, borderline trivial proofs, but perhaps they can come in handy to some high school students or young enthusiasts out there. If you happen to find a mistake in my proofs, or perhaps you have a suggestion how to improve the readability of my texts, be sure to let me know down in the comments section!

See you next time!


Liked this post? Spread the word and follow!

1 comment

Leave a Reply

Your email address will not be published.