What Are Trigonometric Functions Even For? — A Language for Repetition (Waves)

Introduction

Have you ever had a thought like this?

"Why on earth do we study trigonometric functions? sin, cos, the angle addition formula, the double-angle formula… what is all of this even used for?"

I, too, was stuck on this question for a long time. I could memorize the formulas, but I never really understood why these formulas were needed. Since textbooks prove each formula separately and move on, the overall purpose never became visible.

Let me say it in one sentence first.

The purpose of trigonometric functions is to express repetition (waves). Sound, light, electrical signals, the vibration of a swaying building — every "repeating" phenomenon in the world is expressed with sin and cos. The formulas are simply the tools for manipulating those waves.

In this article, we will look at each formula again from the perspective of "why is this needed," and work through every derivation slowly without skipping a single step.

sin(x) is the most basic unit of repetition

Consider a point on the unit circle (a circle of radius 1). If you rotate this point counterclockwise by an angle of \(x\), then the height of the point (its y-coordinate) is \(\sin(x)\).

As you increase \(x\) from 0 to \(2\pi\), the point makes one full loop around the circle. Then \(\sin(x)\) starts at 0, rises to 1, falls down to −1, and returns to 0. And every \(2\pi\), exactly the same shape repeats.

$$ \sin(x + 2\pi) = \sin(x) $$

This is the key point. sin(x) is a function designed to repeat. That is why, when we write down repeating natural phenomena as equations, sin (and cos) appear.

A, ω, φ — three knobs

If we attach three knobs to the basic \(\sin(x)\), we can build any simple repeating signal.

$$ y = A\sin(\omega x + \phi) $$

• \(A\) : amplitude — how big the wave swings up and down
• \(\omega\) : angular frequency — how fast (how densely) it repeats
• \(\phi\) : phase — at what point the wave starts

Stated only in words this stays abstract, so let's plug in some actual numbers.

$$ y = 2\sin\left(3x + \frac{\pi}{2}\right) $$

This expression reads as follows.

• \(A = 2\) : the value swings between \(+2\) and \(-2\). It swings twice as large as the basic \(\sin\) (−1 to +1).
• \(\omega = 3\) : normally \(\sin x\) needs \(x\) to travel \(2\pi\) for one full loop, but here it is \(3x\), so even when \(x\) travels only \(\frac{2\pi}{3}\) it has already made one loop. In other words, it repeats three times as fast. (The period required for one loop \(= \frac{2\pi}{\omega} = \frac{2\pi}{3}\).)
• \(\phi = \frac{\pi}{2}\) : plugging in \(x = 0\) gives \(y = 2\sin\frac{\pi}{2} = 2\times 1 = 2\). So it starts right at the peak (+2) the moment it begins. The phase plays this role of shifting the starting position of the wave.

Once you fix just these three numbers \((A, \omega, \phi)\), you can draw any simple repeating signal. In fact, the musical note "A (A4)" is a wave that vibrates 440 times per second, and we write the air-pressure variation directly in exactly this form, \(y = A\sin(2\pi\cdot 440\cdot t + \phi)\). (Here \(\omega = 2\pi\cdot 440\), in effect.)

Purpose of the formula ①: the angle addition formula — combining two waves

Now let's look at the formula.

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

This formula rewrites "a wave shifted in phase by \(\beta\)" as "a sin·cos combination of the original wave." Why is this needed?

In reality, two waves overlap at the same time. When two speakers emit sound of the same frequency, the two waves combine in space. To know what kind of wave the combined result is, we need the angle addition formula. To add two waves with phases \(\phi_1\) and \(\phi_2\) as equations, we have to expand \(\sin(x + \phi)\), and the very tool for that is the angle addition formula.

In a word: the angle addition formula = the formula that lets us add, as equations, two waves with different phases (different starting times).

The angle addition formula itself comes from computing the coordinates of a point on the unit circle rotated by an angle of \(\alpha + \beta\). In this article we treat this formula as a "tool already in hand," and focus on how the other formulas branch out from it.

Purpose of the formula ②: the double-angle and half-angle formulas — changing the frequency

Pulling the double-angle formula out of the angle addition formula

The double-angle formula is not something to memorize separately; it comes out immediately just by setting \(\beta = \alpha\) in the angle addition formula.

Let's do sin first. Put \(\alpha\) in every \(\beta\) slot of the angle addition formula above.

$$ \sin(2\alpha) = \sin(\alpha + \alpha) = \sin\alpha\cos\alpha + \cos\alpha\sin\alpha = 2\sin\alpha\cos\alpha $$

(The same term \(\sin\alpha\cos\alpha\) appears twice, so it becomes \(2\sin\alpha\cos\alpha\).)

We do cos the same way. But the angle addition formula for cos has a different sign from sin.

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

Putting \(\beta = \alpha\) here gives:

$$ \cos(2\alpha) = \cos(\alpha + \alpha) = \cos\alpha\cos\alpha - \sin\alpha\sin\alpha = \cos^2\alpha - \sin^2\alpha $$

This is the true identity of the \(\cos(2\alpha) = \cos^2\alpha - \sin^2\alpha\) that "looked like it popped out of nowhere." It is not a formula that fell from the sky; it is the result of putting \(\beta = \alpha\) into the cos angle addition formula and simplifying.

The \(\sin(2\alpha)\) built this way repeats twice as fast as \(\sin(\alpha)\). In other words, the double-angle formula is the formula that doubles the frequency.

Pulling the half-angle formula out of the double-angle formula

This time, conversely, let's lower the frequency by half. The starting point is the \(\cos(2\alpha) = \cos^2\alpha - \sin^2\alpha\) we just found.

Here we use the Pythagorean identity \(\sin^2\alpha + \cos^2\alpha = 1\). Rewriting it as \(\cos^2\alpha = 1 - \sin^2\alpha\), we substitute it directly into the \(\cos^2\alpha\) slot of the equation above.

$$ \cos(2\alpha) = \underbrace{(1 - \sin^2\alpha)}_{\cos^2\alpha} - \sin^2\alpha = 1 - 2\sin^2\alpha $$

(\(-\sin^2\alpha\) gathers twice, giving \(-2\sin^2\alpha\).)

Now let's solve this equation for \(\sin^2\alpha\). (This process is exactly what "isolating \(\sin^2\alpha\)" means.) Let's move things one line at a time.

$$ \cos(2\alpha) = 1 - 2\sin^2\alpha $$

Moving \(2\sin^2\alpha\) to the left and \(\cos(2\alpha)\) to the right:

$$ 2\sin^2\alpha = 1 - \cos(2\alpha) $$

Dividing both sides by 2:

$$ \sin^2\alpha = \frac{1 - \cos(2\alpha)}{2} $$

Finally, let's change the angle into nice half-units. Setting \(\alpha = \frac{\theta}{2}\) gives \(2\alpha = \theta\), so putting \(\frac{\theta}{2}\) in the \(\alpha\) slot and \(\theta\) in the \(2\alpha\) slot above:

$$ \sin^2\frac{\theta}{2} = \frac{1 - \cos\theta}{2} $$

Looking at the right side, \(\sin^2\) (a wave with angle \(\theta/2\)) is expressed in terms of \(\cos\theta\) (an angle twice as large, i.e., a wave twice as fast). It is, in effect, a transformation that looks down at half the frequency.

In a word: the double-angle and half-angle formulas = tools for raising or lowering the frequency of a wave.

Purpose of the formula ③: combining — finding the maximum amplitude R

In practice, the most frequently used purpose is this: finding how big a wave results from adding two waves.

$$A_1\sin(x + \phi_1) + A_2\sin(x + \phi_2) = R\sin(x + \phi)$$

Can we tidy this into a single sin wave like the right side? We can. We just expand the left side using the angle addition formula.

Derivation

Apply the angle addition formula to each of the two terms.

$$A_1\sin(x + \phi_1) = A_1\sin x\cos\phi_1 + A_1\cos x\sin\phi_1$$$$A_2\sin(x + \phi_2) = A_2\sin x\cos\phi_2 + A_2\cos x\sin\phi_2$$

After adding the two equations, group the terms attached to \(\sin x\) together, and the terms attached to \(\cos x\) together.

$$= \underbrace{(A_1\cos\phi_1 + A_2\cos\phi_2)}_{a}\,\sin x \;+\; \underbrace{(A_1\sin\phi_1 + A_2\sin\phi_2)}_{b}\,\cos x$$

Now we just need to combine \(a\sin x + b\cos x\) into a single sin. Expanding the target shape \(R\sin(x + \phi)\) with the angle addition formula as well:

$$R\sin(x + \phi) = R\sin x\cos\phi + R\cos x\sin\phi = (R\cos\phi)\sin x + (R\sin\phi)\cos x$$

Matching the coefficients of \(\sin x\) with each other, and \(\cos x\) with each other:

$$R\cos\phi = a, \qquad R\sin\phi = b$$

Finding R — square each of the two equations and add them. (\(\cos^2\phi + \sin^2\phi = 1\) works out cleanly.)

$$R^2\cos^2\phi + R^2\sin^2\phi = a^2 + b^2 \;\Rightarrow\; R^2(\cos^2\phi + \sin^2\phi) = a^2 + b^2 \;\Rightarrow\; R^2 = a^2 + b^2$$$$\boxed{R = \sqrt{a^2 + b^2}} \qquad \text{where}\quad a = A_1\cos\phi_1 + A_2\cos\phi_2,\; b = A_1\sin\phi_1 + A_2\sin\phi_2$$

Finding φ — dividing the two equations makes \(R\) disappear.

$$\frac{R\sin\phi}{R\cos\phi} = \frac{b}{a} \;\Rightarrow\; \tan\phi = \frac{b}{a} \;\Rightarrow\; \phi = \arctan\frac{b}{a}$$

(However, if \(a < 0\), add \(180°\) to land in the correct quadrant.)

Why the hypotenuse, of all things — seeing it as a right triangle

Why does the hypotenuse-shaped \(R = \sqrt{a^2+b^2}\) appear? \(a\sin x\) and \(b\cos x\) can be viewed as the components in two directions that are \(90°\) apart (perpendicular). It is just like going \(a\) horizontally and \(b\) vertically. So the size \(R\) of the combined wave becomes \(\sqrt{a^2 + b^2}\), like the hypotenuse of a right triangle, and the phase \(\phi\) becomes the angle that hypotenuse makes.

Doing it directly with numbers

Words alone don't land, so let's plug in actual numbers. We'll take the two waves like this.

• Wave 1: \(3\sin x\) → amplitude \(A_1 = 3\), phase \(\phi_1 = 0°\)
• Wave 2: \(4\sin(x + 90°)\) → amplitude \(A_2 = 4\), phase \(\phi_2 = 90°\)

First compute \(a, b\). (You only need to know \(\cos 0° = 1,\ \sin 0° = 0,\ \cos 90° = 0,\ \sin 90° = 1\).)

$$a = A_1\cos\phi_1 + A_2\cos\phi_2 = 3\times 1 + 4\times 0 = 3$$$$b = A_1\sin\phi_1 + A_2\sin\phi_2 = 3\times 0 + 4\times 1 = 4$$

Then the maximum amplitude is

$$R = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

The phase is

$$\phi = \arctan\frac{b}{a} = \arctan\frac{4}{3} \approx 53.13°$$

That is,

$$3\sin x + 4\sin(x + 90°) = 5\sin(x + 53.13°)$$

Combining a wave of amplitude 3 with a wave of amplitude 4, offset by \(90°\), produced a wave of amplitude 5, like the hypotenuse of a 3-4-5 right triangle. This is the core picture of combining.

Check — since \(\sin(x + 90°) = \cos x\), the left side is \(3\sin x + 4\cos x\). Expanding the right side with the angle addition formula: \(5(\sin x\cos 53.13° + \cos x\sin 53.13°) = 5(0.6\sin x + 0.8\cos x) = 3\sin x + 4\cos x\). Both sides are exactly equal. (\(\cos 53.13° \approx 0.6,\ \sin 53.13° \approx 0.8\))

Getting a feel for it through special cases

Checking a few special cases shows that this formula is intuitively correct.

• Same direction (\(\phi_1 = \phi_2 = 0°\)): \(a = A_1 + A_2\), \(b = 0\), so \(R = A_1 + A_2\) — the amplitudes simply add.
• Opposite direction (\(\phi_2 - \phi_1 = 180°\)): \(a = A_1 - A_2\), \(b = 0\), so \(R = |A_1 - A_2|\) — the two waves cancel each other out.
• 90° apart: \(R = \sqrt{A_1^2 + A_2^2}\) — they add up by the Pythagorean theorem, just like the 3-4-5 example above.

Try it yourself

Below, adjust the amplitude (A) and phase (φ) of the two waves and watch how the combined wave (green) changes. The preset buttons let you instantly compare the special cases.

• "Same direction": the two waves share the same phase → the amplitudes simply add.
• "Opposite direction": phase difference 180° → they cancel, so R ≈ 0.
• "90° apart": phase difference 90° → R = √(A₁² + A₂²), like Pythagoras.
• "General combination": freely combine A₁, A₂, φ₁, φ₂. (Try plugging in A₁=3, A₂=4, phase difference 90° from the main text above and check whether R=5 comes out.)

Combining two waves — try it yourself

If you slowly raise the phase difference (Δφ) from 0° to 180°, you can see R decrease smoothly from A₁+A₂ to |A₁−A₂|.

Key takeaways

Formula	Purpose
\(\sin(x)\) itself	the basic unit of repetition (waves)
angle addition formula \(\sin(\alpha+\beta)\)	combining, as equations, two waves with different phases
double-angle formula \(\sin(2\alpha)\)	doubling the wave frequency
half-angle formula \(\sin^2\tfrac{\theta}{2}\)	halving the wave frequency
combining \(R\sin(x+\phi)\)	computing the maximum amplitude and phase of two waves

Collecting just the key formulas again:

• Double-angle: \(\sin(2\alpha) = 2\sin\alpha\cos\alpha\), \(\cos(2\alpha) = \cos^2\alpha - \sin^2\alpha = 1 - 2\sin^2\alpha\)
• Half-angle: \(\sin^2\dfrac{\theta}{2} = \dfrac{1 - \cos\theta}{2}\)
• Combining: \(R = \sqrt{a^2 + b^2}\), \(\tan\phi = \dfrac{b}{a}\), where \(a = A_1\cos\phi_1 + A_2\cos\phi_2\), \(b = A_1\sin\phi_1 + A_2\sin\phi_2\)
• Special cases: same direction gives \(R = A_1 + A_2\), opposite direction gives \(R = |A_1 - A_2|\), 90° apart gives \(R = \sqrt{A_1^2 + A_2^2}\)

Studying this with AI

When I first encountered this concept, the hardest part was "why the formula is needed." Textbooks show only the derivation; you don't find out where it is used until much later. If you ask an AI, "explain with an example how the angle addition formula is used in wave superposition," it shows you the purpose first, which makes it great for learning in the reverse direction.

That said, it's important to make a habit of cross-verifying the figures the AI gives you (especially computed results) on your own. If you pair it with a tool like the interactive above, which actually computes and displays the values, you can confirm the accuracy of the AI's explanation right then and there.

Wrapping up

The reason the trigonometric formulas felt like they were drifting apart, each on its own, was that their purpose was not visible. Once you know the purpose, the formulas become a single story.

1. sin(x) → the basic language of repetition
2. the angle addition formula → the grammar for combining two waves
3. double-angle / half-angle → the transformations that adjust frequency
4. combining R → computing the size of the combined wave

It won't make the memorization disappear, but I think it will make far more sense if, each time you meet a formula, "the role it plays in waves" comes to mind.

Related reading — Why does the arctangent appear? · Why does Cramer's rule have that shape? · Why are direction cosines a "direction"? · Check the waveform: Desmos Graphing Calculator · Wolfram Alpha