The word "derivative" literally means "dividing into small parts," but its essence is "a magnifying glass that freezes a moving thing at an instant, like taking a photograph, so we can discover how it is changing (its speed) at that moment."

🍎 1. Why did Newton think of derivatives? (The birth of calculus)

In 1665, while staying at his childhood home on an apple farm because of the plague, twenty-something Isaac Newton found himself deeply pondering the "speed of moving things" as he watched the Moon and apples.

• Mathematics of the past (average speed): "If you traveled 400 km in 4 hours, that's 100 km/h!"
• Newton's question (instantaneous speed): "But a car doesn't travel at 100 km/h the whole time, does it? How do you calculate its speed during the very 'instant' (0.00001 seconds) when a speed camera catches it?"

Existing mathematics could not determine the "state at just one moment" of an object whose speed changed in real time, so Newton invented a new mathematical tool himself: the derivative.

---

🔍 2. Visualizing the idea: from average speed to instantaneous speed

The key idea behind differentiation is to measure speed by splitting the time interval into pieces that get arbitrarily close to zero (infinitely small), such as $1$ second $\rightarrow$ $0.1$ seconds $\rightarrow$ $0.0000001$ seconds.

graph LR
    subgraph Past ["Mathematics of the past (average speed)"]
        A["When the time interval is large<br/>(e.g., travel over 1 hour)"] --> B("A line connecting two points<br/>(a secant)")
    end

    subgraph Future ["The magic of derivatives (instantaneous speed)"]
        C["Narrow the time interval endlessly toward zero<br/>(e.g., 0.0000001 seconds)"] --> D("A line that touches and passes through one point<br/>(a tangent)")
    end

    B -.->|"Send the time interval (h) to 0"| C

    style B fill:#f9f2f4,stroke:#d3b8ae,stroke-width:2px
    style D fill:#e0f2f1,stroke:#80cbc4,stroke-width:2px

---

✂️ 3. How do we calculate a derivative? (The magic formula)

Suppose the distance an object travels is the square of time ($x$). In mathematical notation, this is $y = x^2$. The magic spell for differentiating this is a very simple two-step process.

1. Bring it down from above: Bring the little number at the upper right (the exponent) down in front and multiply by it.
2. Subtract one: Subtract $1$ from the little number that was there originally.

$$y = x^2$$

• Bring the number $2$ down in front, and subtract $1$ from the original exponent.
• Result: $2x$

Now just put the time ($x$) you want to know into this expression, $2x$, and you get the "instantaneous speed" at that time. (For example: the instantaneous speed at $3$ seconds is $2 \times 3 = 6$.)

---

🕵️‍♂️ 4. The secret behind the magic formula (proof of the principle)

Mathematicians did not simply cast a spell and tell everyone to memorize it. If we call the fleeting time interval $h$, then calculate average speed and make $h$ disappear by taking it to $0$, the same rule always emerges.

graph TD
    Step1["Step 1: Set up the speed expression<br/>Change in distance / time interval (h)"] --> Step2["Step 2: Expand the parentheses<br/>(apply the multiplication identity)"]
    Step2 --> Step3["Step 3: Simplify the fraction<br/>(cancel h in the denominator and numerator)"]
    Step3 --> Step4["Step 4: Make h vanish to 0<br/>(the fleeting instant)"]
    Step4 --> Result["Derive the result<br/>2x remains without complicated calculations!"]

    style Result fill:#e3f2fd,stroke:#1976d2,stroke-width:2px

Step 1: Set up a speed expression using distance traveled and time taken * Current time: $x$; after a fleeting moment has passed: $x + h$ * Time elapsed: $h$; distance traveled: $(x + h)^2 - x^2$

$$\text{Average speed} = \frac{(x + h)^2 - x^2}{h}$$

Step 2: Expand the parentheses $$(x + h)^2 = x^2 + 2xh + h^2$$ Substituting this into the expression above makes the $x^2$ terms at the beginning and end cancel.

$$\frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h}$$

Step 3: Simplify the fraction Divide the numerator and denominator by $h$.

$$\frac{h(2x + h)}{h} = 2x + h$$

Step 4: Make the time interval ($h$) vanish to $0$ Since we want the speed at an instant, we think of the interval $h$ as being $0$.

$$2x + 0 = 2x$$

Thus, the perfect rule "bring down the exponent and subtract one", which lets us skip a complicated calculation, has been proven.

---

🚀 5. Where are derivatives used?

Derivatives are used whenever we analyze "something that is changing."

• 🚗 Car speedometer: The current speed indicated by the needle is itself the result of differentiation.
• 📈 Predicting stocks and cryptocurrencies: The slope tells us whether prices are "on an upward trend" or "starting to turn downward."
• 🤖 Artificial intelligence (AI) deep learning: AI uses derivatives to find the direction in which it can reduce errors the fastest.
• 🎢 Roller-coaster design: They calculate the safest and most thrilling angles and speeds that will keep riders from flying off.

2. Integrals - derivatives' best friend

The word "integral" uses the characters for "accumulating" and "dividing." In other words, it is a "piggy bank that carefully piles up tiny, divided instants again to find the original whole (an area or volume)."

🍎 1. Did Newton also invent integrals? (The fundamental theorem of calculus)

Yes, that's right! While studying derivatives, Newton also systematized the reverse process: integration.

• Differentiation: Differentiate distance $\rightarrow$ instantaneous speed.
• Integration: Gather and accumulate the instantaneous speeds (integrate them) $\rightarrow$ recover the distance traveled.

Newton uncovered the astonishing secret that differentiation and integration are essentially completely inverse operations (two sides of the same coin). In mathematics, this is called the "fundamental theorem of calculus." (Incidentally, the S-shaped integral symbol $\int$ (integral) that we use was created by Newton's rival, the German mathematician Leibniz.)

---

🔍 2. Visualizing the idea: the relationship between differentiation and integration

If differentiation finds a graph's "slope," integration finds the "area" under the graph.

graph LR
    subgraph Circle ["The cycle of calculus"]
        A["📍 Position / distance<br/>(e.g., y = x²)"] -->|"Differentiate ✂️<br/>(find the slope)"| B["🚀 Instantaneous speed<br/>(e.g., y = 2x)"]
        B -->|"Integrate 🧱<br/>(find the area)"| A
    end

    style A fill:#e3f2fd,stroke:#1976d2,stroke-width:2px
    style B fill:#fbe9e7,stroke:#d84315,stroke-width:2px

---

🪄 3. How do we calculate an integral? (The reverse magic formula)

When differentiating, we "bring down the exponent and subtract 1," right? Integration goes back in time, so we do exactly the reverse.

"Add 1, then divide by that number!"

Let's integrate the speed expression we found earlier, $y = 2x$, and return to the original distance expression ($x^2$). ($2x$ is really $2x^1$, with a hidden exponent of 1.)

1. Add one: Add $1$ to the original exponent on $x$, which is $1$. $\rightarrow 1 + 1 = 2$ (the new exponent is $2$)
2. Divide: Divide the number $2$ in front by the new exponent we just found ($2$). $\rightarrow 2 \div 2 = 1$

$$ \int 2x \, dx $$

• The coefficient becomes $1$, and the exponent of $x$ becomes $2$.
• Result: $x^2$

🎉 Ta-da! We have perfectly returned to the original form, $x^2$, from before differentiation! (Note: In actual mathematics, we may not know the starting position, so we add an unknown constant $C$ at the end of an integral and write $x^2 + C$.)

---

🕵️‍♂️ 4. The secret behind the integral formula (working backward)

How was the integration rule ("add 1 to the exponent, then divide by that number") created? Rather than calculate integrals from scratch in a complicated way, mathematicians traced the expression backward like detectives, using Newton's discovery that "if you reverse differentiation, you get integration."

Goal: Find the integral of $x^2$! (Let's call the result $\square$ for now.)

Step 1: Recall the relationship between differentiation and integration If we differentiate the integral result, $\square$, we should get the original form, $x^2$.

$\square \rightarrow \text{differentiate} \rightarrow x^2$

Step 2: Deduce the degree (exponent) Differentiation reduces the exponent by 1. So what does it mean if the result is the second power ($x^2$)? The original $\square$ must have contained the third power ($x^3$).

$(x^3) \rightarrow \text{differentiate} \rightarrow 3x^2$

Step 3: Get rid of the unwelcome number When we differentiated $x^3$, we got the $x^2$ we wanted, but an unwelcome $3$ appeared in front. ($3x^2$) To get rid of this $3$, the original expression must have been multiplied by $\frac{1}{3}$, so the factors cancel each other out, right?

$(\frac{1}{3}x^3) \rightarrow \text{differentiate} \rightarrow 3 \times \frac{1}{3}x^2 = x^2$

Step 4: Complete the formula We confirmed that we get exactly $x^2$! That means the integral of $x^2$ is $\frac{1}{3}x^3$. Through this process, mathematicians completed a universal reverse formula that works for every number.

1. Differentiation reduces the exponent by 1. $\rightarrow$ Integration adds 1 to the exponent. ($2 \rightarrow 3$)
2. Differentiation brings the exponent down in front as a multiplier. $\rightarrow$ Integration divides by the new exponent. ($\div 3$)

graph LR
    A["Integral result (unknown)<br/>?"] -->|"Differentiate"| B["Current state<br/>x²"]

    C["Deduction 1: Raise the exponent<br/>x³"] -->|"Differentiate"| D["3x² (3 remains)"]

    E["Deduction 2: Divide by the number<br/>1/3 x³"] -->|"Differentiate"| F["x² (Perfect!)"]

    style E fill:#e8f5e9,stroke:#2e7d32,stroke-width:2px
    style F fill:#e8f5e9,stroke:#2e7d32,stroke-width:2px

🧱 6. Visualization: Why use an integration formula at all? (Riemann sums)

Let's assume we don't know the integration formula. To find the area under a curve, we would have to divide the area below it into a great many tiny rectangles and add up every one of their areas, just as mathematicians did in the past. (In mathematics, this is called a "Riemann sum.")

But splitting these rectangles into infinitely small pieces and adding them is nearly impossible for a person to do. That's why the "magical reverse formula (add 1 to the exponent and divide)" we just proved is such an incredible invention that changed the world!

graph LR
    subgraph "The old way (Riemann-sum drudgery)"
        A["Split into 10 rectangles<br/>(large protruding errors)"] --> B["Split into 10,000 rectangles<br/>(exhausted from calculating)"]
    end

    subgraph "Newton's invention (integration formula)"
        C["Put a number into the formula<br/>(get the exact area in 1 second!)"]
    end

    B -.->|"Split infinitely (Limit)"| C

    style A fill:#fff3e0,stroke:#e65100,stroke-width:2px
    style B fill:#ffe0b2,stroke:#ef6c00,stroke-width:2px
    style C fill:#e8f5e9,stroke:#2e7d32,stroke-width:2px

• Riemann sum (past): "Rectangle area number 1 + number 2 + ... a million times... Oh, how will I ever finish adding them all?"
• Integration formula (present): "Just plug a number into the expression, and the area split into infinitely tiny pieces appears in one second!"

In short, instead of laboriously adding up the areas of thousands of rectangles, the integration formula our predecessors developed lets us find the exact area and volume of a curve in just one second.

🚀 5. Where are integrals used?

If differentiation looks at a "changing state," integration is used whenever we want to find an "accumulated total."

• 🏥 CT scans (medicine): X-rays are taken by slicing the body thinly (into cross-sections) from many angles, as if differentiating it; then a computer "integrates" them again to reconstruct 3D organs.
• 🚀 Spacecraft trajectory calculations: By carefully integrating a spacecraft's constantly changing speed data, we calculate its exact current position, including how many kilometers it is from Earth.
• 🏛️ Architecture and civil engineering: Integration, which finds area by dividing curves into tiny pieces, is essential for finding the total area and volume of a rounded roof (dome) or an irregularly shaped lake.
• 🔋 Battery-level display: The current flowing at each instant as a smartphone battery discharges is integrated (accumulated) to calculate what percentage of the battery remains.

3. Calculus in the real world: How do we find an equation?

So far, we have learned calculus assuming that we have a perfect equation, $y = x^2$. But a real car moves very irregularly, braking and accelerating. So how on earth do we find an equation and do calculus in the real world?

🍎 1. Where did $y = x^2$ come from? (Ideal physical laws)

The $y = x^2$ in our first example is actually an ideal model that greatly simplifies the "law of free fall" Newton learned from watching an apple fall, or a situation in which an object "continues accelerating under a constant force."

It is closer to a rocket flying through space without air resistance, continuously shooting a steady flame out of its tail, than to a real car. In other words, it keeps only the framework needed to make a complex reality easy to calculate.

---

📡 2. How can we calculate irregular motion in the real world?

In the irregular real world, where cars repeatedly start and stop, we mainly use two methods.

Method A: Collect data and force a best-fit equation (regression analysis)

There is no perfect equation to start with. But we do have sensors (GPS, speedometers, and so on).

1. Collect data: A GPS installed in a car records its position as points (data) every 0.1 seconds.
2. Plot the points: When these thousands of points are plotted on a computer screen, they form an irregular, winding line.
3. Draw a trend line (curve fitting): Computer AI or statistical software finds a "similar equation" that connects these points as well as possible (with the least error). (For example: "Hmm, this car's motion is 98% similar to the equation $y = 0.3x^3 - 1.2x^2 + 5x + 2$!")
4. Apply calculus: The computer differentiates this complicated equation it found by approximation to determine the instantaneous speed at a given point.

Method B: Forget equations and calculate directly from points (numerical analysis)

This is the method computers and AI use most often today. There is no need to go to the trouble of finding a complicated equation: we can use the data points recorded every 0.01 seconds to perform differentiation and integration directly.

• Real-world differentiation (numerical differentiation): Take just two position points, one recorded at 1.00 seconds and one at 1.01 seconds, and find the slope (speed) between them. Since the time interval is only 0.01 seconds—a fleeting instant—this is effectively the instantaneous speed (the derivative).
• Real-world integration (numerical integration): Treat the speed data arriving from sensors every 0.01 seconds as rectangles. A program such as Excel or Python adds the areas of tens of thousands of these rectangles in real time to find the total distance traveled. (The computer does the "Riemann-sum drudgery" we saw earlier in one second.)

---

🎯 Conclusion: The calculus process in the real world

1. The past (math textbooks): First provide a perfect equation ($y = x^2$), as if handed down by a god, and ask students to differentiate it.
2. The present (programming/AI): First collect countless data points using sensors, then use a computer to perform numerical differentiation/integration in real time to control a self-driving car.

The reason we practice calculus in math class using perfect equations such as $y=x^2$ is that Newton's discovery provides the essential underlying principles computers use when they perform numerical calculus on points!

4. Straight line vs. curve: Which is faster? (The brachistochrone problem)

When rolling a ball down from a high place, which reaches the destination first: a straight slide that goes directly there, or a curved slide that bends roundly downward?

In 1696, Swiss mathematical genius Johann Bernoulli provocatively posed this "brachistochrone problem" to mathematicians throughout Europe.

🏁 1. Intuition betrayed: the shortest distance is not always the shortest time!

Common sense suggests that the "straight line," the shortest distance, would be the fastest. But the answer is the "cycloid," a curve that dips down in a rounded shape. (It looks similar to an ellipse, but is mathematically a different curve.)

• Straight-line path: It is the shortest distance, but acceleration builds slowly, so the speed remains mediocre throughout the descent.
• Cycloid path: It is longer than a straight line. But it drops steeply right from the start, building tremendous acceleration (initial speed), so it races through the remaining long distance in the blink of an eye and arrives first overall.

graph LR
    subgraph "Shortest distance vs. shortest time"
        A((Ball)) -->|Straight path<br/>Shorter, but slower| B[Destination]
        A -->|Cycloid path<br/>Longer, but fastest thanks to an explosive early acceleration!| B
    end

    style A fill:#ffcc80,stroke:#e65100,stroke-width:2px
    style B fill:#c8e6c9,stroke:#2e7d32,stroke-width:2px

---

🦁 2. Newton's overnight solution: "The lion is known by its claw mark"

Johann Bernoulli sent a taunting letter to the greatest mathematicians of the time (including Leibniz and l'Hôpital), challenging them to solve the problem within six months.

In his fifties, Isaac Newton, exhausted after working all day as Master of the Royal Mint and fighting counterfeiters, came home and discovered this math problem among his mail at 4 p.m.

Angered by this challenge to his mathematical pride, Newton skipped dinner and worked on the calculations all night, solving this devilishly difficult problem perfectly in just one night (by 4 a.m.).

Newton sent the solution to Europe anonymously, hiding his name. But when Johann Bernoulli saw the overwhelmingly powerful and elegant calculus in the solution, he exclaimed:

"Tanquam ex ungue leonem." "I know it is a lion (Newton) from its claw mark alone."

---

🕵️‍♂️ 3. How did Newton find this equation? (The birth of the calculus of variations)

Ordinary differentiation is a technique for finding the "lowest point (minimum)" on a given equation ($y=x^2$).

But this problem was on another level. No equation was given; the challenge was to "find the perfect curve equation itself that minimizes the time."

Geniuses such as Newton and Bernoulli created new mathematics to solve it. 1. Use the principle of light: Bernoulli combined physics and calculus with the optical law that "light always bends by choosing the fastest path" (Fermat's principle). 2. Calculus that finds functions: They upgraded differentiation one step further, differentiating not numbers but "the function itself" to work backward and discover the optimal path curve (the cycloid).

This event led to the birth of "the calculus of variations," a very important branch of mathematics that forms the foundation of physics and artificial intelligence (AI). Even the algorithms AI uses today to find optimal answers are descendants of this calculus of variations!

5. Differentiation that finds functions: the calculus of variations

The new mathematics Newton and the great mathematicians of his time invented to find the "curve of fastest descent (the cycloid)" was the calculus of variations. The name sounds difficult, but the idea is ingenious and satisfying.

🎯 1. Ordinary differentiation vs. the calculus of variations (What's the difference?)

The biggest difference is "what we are trying to find."

• Ordinary differentiation (mathematics for finding a point): Given a fixed mountain (an equation, $y=x^2$), find the lowest valley "point (coordinate value)" on that mountain. The answer is always a "number" such as $x=2$.
• The calculus of variations (mathematics for finding a curve): There is no mountain to start with. It is a quest that says, "Create the shape of the mountain yourself so you can get from A to B as quickly as possible!" The answer is not a number, but the "function (equation) itself," such as a cycloid.

🎸 2. How did Newton differentiate a "curve"? (The perturbation technique)

To discover a perfect curve in empty space, mathematicians came up with the brilliant idea of "variation." It is a bit like plucking a guitar string.

1. Draw a hypothetical curve: Start by drawing one curve that might be the answer.
2. Give it a little shake (variation): Push a tiny part of the curve slightly up or down, as if plucking it with a finger. (This is a variation.)
3. Check the time: If the time it takes gets shorter when you shake the curve a little, then the previous curve was not the answer! It means there is a faster route that takes less time.
4. The condition for the answer: The perfect curve is the one whose travel time only ever increases, no matter how you shake or distort the guitar string! That curve is the optimal answer (the cycloid).

graph TD
    subgraph "Ordinary differentiation (traditional mathematics)"
        A["Find the lowest 'point'<br/>on a fixed graph"] --> B["Result: a number (x=3)"]
    end

    subgraph "The calculus of variations (Newton's invention)"
        C["Shake an arbitrary curve up and down<br/>to find the optimal 'shape'"] --> D["Result: a function (y = cycloid equation)"]
    end

    style A fill:#f3e5f5,stroke:#8e24aa,stroke-width:2px
    style C fill:#e8eaf6,stroke:#3f51b5,stroke-width:2px

🚀 3. Where is the calculus of variations used in modern society?

It was originally created to calculate how long a rolling ball takes, but today the calculus of variations is one of the most powerful tools that moves the world.

• 🚗 Navigation and self-driving cars: "Among tens of thousands of routes from Gangnam Station to Haeundae Beach in Busan, find the optimal 'route (function)' that uses the least time and fuel!"
• 🤖 Artificial intelligence (AI) deep learning: When AI learns: "Among countless error curves, find the perfect 'learning path' that reaches the right answer most quickly and accurately!" (Gradient descent, used for this purpose, is built on the philosophy of the calculus of variations.)
• ⚛️ Quantum mechanics and the universe: When light and electrons travel, they independently choose the path in the universe that uses the least energy ("the principle of least action"). The very laws of the universe operate through the calculus of variations.

Conclusion: Newton's overnight claw mark (the calculus of variations) remains firmly alive today, hundreds of years later, as a core principle that computers and artificial intelligence use to find optimal answers.

6. The geniuses' clever trick: How did they find the equation? (The principle of light)

How did Newton and Johann Bernoulli find a perfect curve called a cycloid out of thin air? Instead of complicated mathematical formulas, they got a hint from "light," nature's greatest runner.

🏖️ 1. The lifeguard's dilemma (Fermat's principle)

It had been discovered that light always chooses and travels along the "path that takes the least time (the path of least time)" in the universe. (This is called Fermat's principle of least time.)

Imagine a lifeguard on the beach going to rescue someone who has fallen into the sea. * Sand (running): The speed is very fast. * Sea (swimming): The speed is very slow.

What happens if the lifeguard runs straight toward the destination? They have to swim through the slow sea for too long, so they arrive later. So the fastest way to arrive is to run farther over the sand, where running is fast, and bend the route so that the slow swim across the sea is as short as possible! (This is the law of refraction that makes a straw submerged in water look bent.)

graph LR
    subgraph "The fastest route is not a straight line!"
        A["🏃 Lifeguard (sand: fast)"] -->|"Straight route (takes longer)"| B["🌊 Sea (swimming: slow)"]
        A -->|"Run farther on the sand, then turn into the water<br/>(arrive fastest!)"| C["🏊 Person in the water"]
        B -.-> C
    end

    style A fill:#fff3e0,stroke:#e65100,stroke-width:2px
    style C fill:#e1f5fe,stroke:#0288d1,stroke-width:2px

📉 2. Combining gravity and light (Newton's magic)

Let's return to the brachistochrone problem. As a ball falls downward, gravity makes it move faster and faster.

The geniuses thought of this situation as exactly analogous to the refraction of light. * Space above: A region where speed is slow (like the sea, where you have to swim) * Space below: A region where speed is extremely fast (like the sand, where it is easy to run)

"How would light pass through this space?" Light would try to escape the slow upper region by dropping straight down as quickly as possible. In the fast lower region, it would move gently, enjoying that fast speed for a long time even if the distance were longer.

🏆 3. The answer: the birth of the cycloid

They divided space into tens of thousands of layers and calculated how light bends as its speed increases from layer to layer. The result was an elegant curve that drops steeply at first and becomes gentler as it goes along. When they translated this path of light into mathematical symbols, the equation that emerged was exactly the "cycloid" curve!

In the end, Newton and the mathematicians mathematically proved the philosophical belief that "nature (light) already knows the answer," leaving behind an astonishing achievement that became the foundation of modern physics and quantum mechanics.

7. How much does light bend? (Snell's law)

The amount light bends when it enters one material from another is determined by the "difference in speed" between the two materials. This relationship is expressed mathematically by Snell's law.

📐 1. Snell's law formula

When light enters a medium with refractive index $n_2$ from one with refractive index $n_1$, there is a fixed relationship between the incoming angle ($\theta_1$) and the refracted angle ($\theta_2$).

$$n_1 \times \sin(\theta_1) = n_2 \times \sin(\theta_2)$$

• $n$ (refractive index): A value indicating how much light slows down inside a material (vacuum = 1, air ≒ 1, water ≒ 1.33)
• $\theta$ (angle): The angle of the light measured from the perpendicular line (the normal)
• $\sin$ (sine): A trigonometric function that represents the ratio of height for a given angle

🚗 2. Why does it bend? (The wheel analogy)

Imagine a car driving along a paved road and entering a muddy area at an angle. 1. The wheel that reaches the mud first slows down significantly. 2. The wheel on the other side is still moving quickly on the road. 3. This difference in speed makes the car turn sharply toward the mud.

Light bends toward the material where its speed is slower according to the very same principle.

graph TD
    subgraph "Path taken by light"
        A["🔦 Incoming light (air: fast)"] -->|"Angle of incidence θ1"| B{Boundary}
        B -->|"Angle of refraction θ2"| C["🌊 Light travels on (water: slow)"]
    end

    B -.->|"Condition for total internal reflection"| D["Light cannot pass through and bounces back<br/>(at angles greater than the critical angle)"]

    style B fill:#e1f5fe,stroke:#01579b,stroke-width:2px
    style D fill:#ffebee,stroke:#c62828,stroke-width:2px

---

8. Coming next: trigonometric functions, the language of calculus

To explore calculus and physics in depth, we need the tool called trigonometric functions, such as $\sin$ (sine), which appeared in Snell's law.

• What are trigonometric functions? Magical functions that turn the "angles" of a triangle into "ratios of lengths."
• Why study them? They are the only language that explains all the world's "repeating waves," from ocean waves, sound, light, and electricity.

In the next chapter, we will explore in an easy-to-understand way how trigonometric functions are used in calculus and why we should study them.