# Exploring the Astonishing Properties of Trig Functions

• Char. Limit
In summary, the conversation discusses various properties and theorems of trigonometric functions, including the Law of Cosines, the Pythagorean Theorem, and integrals involving trigonometric functions. The conversation also mentions interesting equations and series involving trigonometric functions, such as the well-known relationship between sine and cosine and the surprising product involving secants and pi. The conversation ends with a discussion on the difficulty of integrating certain cosine products and the use of identities to solve them.
Char. Limit
Gold Member
...are fascinating. At least I think so. Sine and cosine are the additive inverses of their respective second derivatives, for example. Astonishing!

Are there any other startling properties of trig functions (not inverse trigs) that would just blow my mind?

Somewhere in the beautiful scale between $$a^2+b^2=c^2$$ and $$e^{i \pi }+1=0$$ is what I'm looking for, I guess.

Check out the Law of Cosines, aka the Pythagorean Theorem for non-right triangles.

Well the first property you mentioned is related to a well known theorem. If f is some function that has a second derivative everywhere, and f'' + f = 0, f(0) = a, f'(0) = b. Then f = b*sin + a*cos.

Also, it's well-known that

$$\int_{-\infty}^{\infty}\frac{\sin x}{x}\,dx = \pi.$$

You should try proving the first, since it only requires differentiation. The second is much harder to prove (demonstrating convergence is even a bit tricky), and perhaps less of what you were looking for.

One of my personal favorites is

$$2 \prod_{k=1}^{\infty} sec (\frac{\pi}{2^{k+1}}) = \pi$$

or even the elegant

$$\int_{0}^{\pi} cos^n (\theta) cos (n \theta) d\theta = \frac{\pi}{2^n}, n = 0, 1, 2, 3...$$

Nonsense, I'm looking for anything particularly interesting.

On the first theorem presented, I suppose it is intuitive. Since at x=0, a constant multiplied by the sine of x is also 0. Thus, the two constants "switch" in $$f^{(n)}(0)$$ as n increases, changing sign each time they appear.

Right?

Of course, for the second equation, the proof is obvious!

*leaves room to determine proof*

Edit: Gah, took too long to post. Is the giant pi symbol used for the same thing as the giant sigma in summation?

Also, I'm currently trying to remember how to integrate cosine products. I'll try by parts...

Char. Limit said:
Nonsense, I'm looking for anything particularly interesting.

On the first theorem presented, I suppose it is intuitive. Since at x=0, a constant multiplied by the sine of x is also 0. Thus, the two constants "switch" in $$f^{(n)}(0)$$ as n increases, changing sign each time they appear.

Right?

Of course, for the second equation, the proof is obvious!

*leaves room to determine proof*

Edit: Gah, took too long to post. Is the giant pi symbol used for the same thing as the giant sigma in summation?

Also, I'm currently trying to remember how to integrate cosine products. I'll try by parts...

The giant pi symbol is "product" as sigma is "sum".

l'Hôpital said:
The giant pi symbol is "product" as sigma is "sum".

I see.

Well, for the cosine product function, integration by parts won't work (every two applications, the non-integral part cancels out), I'm pretty sure integration by substitution wouldn't work, and with my limited repertoire (is that how you spell it?), I'm out of options. Same with the sine of x over x function. Infinite products of secants... I doubt it's currently in my ability to understand what's going on, much less prove it.

Und so falle ich.

I am impressed, though. How many series equal some multiple, power, or multiplied power of pi anyways?

Char. Limit said:
I see.

Well, for the cosine product function, integration by parts won't work (every two applications, the non-integral part cancels out), I'm pretty sure integration by substitution wouldn't work, and with my limited repertoire (is that how you spell it?), I'm out of options. Same with the sine of x over x function. Infinite products of secants... I doubt it's currently in my ability to understand what's going on, much less prove it.

Und so falle ich.

I am impressed, though. How many series equal some multiple, power, or multiplied power of pi anyways?

The cosine integral is tricky. The product? Not so much.

Hint: Consider $$sin 2\theta = 2 sin \theta cos \theta$$

l'Hôpital said:
The cosine integral is tricky. The product? Not so much.

Hint: Consider $$sin 2\theta = 2 sin \theta cos \theta$$

I'm considering it...

Is there a sin(a)/sin(b)= identity anywhere? There's a sin(a)sin(b) identity in this book, but I don't need that...

## 1. What are trigonometric functions?

Trigonometric functions, also known as trig functions, are mathematical functions that relate the angles of a right triangle to the lengths of its sides. The most commonly used trig functions are sine, cosine, and tangent.

## 2. How are trig functions used in science?

Trig functions are used in many scientific fields, including physics, engineering, and astronomy. They are used to model and analyze periodic phenomena, such as the motion of waves, vibrations, and oscillations.

## 3. What are some real-world applications of trig functions?

Trig functions have many practical applications, such as calculating the height of buildings, determining the distance between two objects, and analyzing the trajectory of a projectile. They are also used in navigation, surveying, and GPS systems.

## 4. How do trig functions relate to the unit circle?

The unit circle is a circle with a radius of 1 and is centered at the origin on a Cartesian plane. Trig functions are defined as the ratios of the sides of a right triangle in relation to the unit circle. This allows for a visual representation of the trig functions and their properties.

## 5. What are the key properties of trig functions?

The key properties of trig functions include periodicity, symmetry, and the relationships between the different functions. They also have important identities, such as the Pythagorean identity, which is used to simplify expressions involving trig functions.

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