Introduction: The Language of Rotation

In the last chapter (3.1), we explored how some things in life happen in repeating cycles, like tides or heartbeats. But how do we actually calculate where something is in its cycle? That’s where Sine, Cosine, and Tangent come in! These aren't just buttons on your calculator; they are functions that tell us exactly where a point is on a circle based on the angle of rotation. Think of them as the "GPS coordinates" for anything that moves in a circle or a wave.

1. Angles in Standard Position

Before we define our three functions, we need a consistent way to draw angles. In AP Precalculus, we always use standard position:

  • The vertex (the corner of the angle) is at the origin \((0, 0)\).
  • The initial side (where the angle starts) sits on the positive \(x\)-axis.
  • The terminal side (where the angle ends) rotates counter-clockwise for positive angles and clockwise for negative angles.

Imagine a clock hand starting at 3:00 and sweeping upward. The distance from the origin to any point \((x, y)\) on that hand is the radius, which we call \(r\). We can find \(r\) using the Pythagorean Theorem: \(r = \sqrt{x^2 + y^2}\).

Quick Review: Remember that in this course, we assume angles are in radians unless you see a degree symbol (\(^\circ\)). A full circle is \(2\pi\) radians.

2. Defining the "Big Three" Functions

If we pick a point \((x, y)\) on the terminal side of an angle \(\theta\), and that point is a distance \(r\) from the origin, we define our functions as ratios:

The Cosine Function (\(\cos \theta\))

The cosine of an angle is the ratio of the \(x\)-coordinate to the radius.
\(\cos(\theta) = \frac{x}{r}\)

The Sine Function (\(\sin \theta\))

The sine of an angle is the ratio of the \(y\)-coordinate to the radius.
\(\sin(\theta) = \frac{y}{r}\)

The Tangent Function (\(\tan \theta\))

The tangent of an angle is the ratio of the \(y\)-coordinate to the \(x\)-coordinate.
\(\tan(\theta) = \frac{y}{x}\) (where \(x \neq 0\))

Analogy: Imagine you are walking around a circular track. Cosine tracks how far Left or Right you are from the center. Sine tracks how far Up or Down you are from the center. Tangent tracks the "steepness" or slope of your path from the center to your current spot.

Key Takeaway:

Cosine is associated with \(x\), Sine is associated with \(y\), and Tangent is the ratio \(\frac{y}{x}\).

3. The Magic of the Unit Circle

Everything gets much easier when we set the radius to \(r = 1\). This is called the Unit Circle. Because dividing by \(1\) doesn't change a number, the definitions become incredibly simple:

  • \(\cos(\theta) = x\)
  • \(\sin(\theta) = y\)
  • \(\tan(\theta) = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}\)

On the Unit Circle, every point on the edge is literally \((\cos \theta, \sin \theta)\). If you know the coordinates of a point on the Unit Circle, you already know the sine and cosine of that angle!

Did you know? The word "sine" comes from a series of mistranslations of the Sanskrit word jya (bowstring). It eventually became the Latin sinus, meaning "bay" or "curve."

4. Signs and Quadrants (Where is it positive?)

Depending on which quadrant the terminal side of the angle falls in, \(x\) and \(y\) can be positive or negative. This affects the signs of our trig functions:

  • Quadrant I (Top Right): Both \(x\) and \(y\) are positive. All functions (Sin, Cos, Tan) are positive.
  • Quadrant II (Top Left): \(x\) is negative, \(y\) is positive. Only Sine is positive.
  • Quadrant III (Bottom Left): Both \(x\) and \(y\) are negative. Only Tangent is positive (because a negative divided by a negative is a positive!).
  • Quadrant IV (Bottom Right): \(x\) is positive, \(y\) is negative. Only Cosine is positive.

Memory Aid: Use the phrase "All Students Take Calculus" to remember which functions are positive in Quadrants I, II, III, and IV respectively.

5. Tangent and Slope

One of the most important connections in AP Precalculus is that tangent represents the slope of the line containing the terminal side of the angle. Since slope is \(\frac{\text{rise}}{\text{run}}\) or \(\frac{\Delta y}{\Delta x}\), and \(\tan(\theta) = \frac{y}{x}\), they are the same thing!

Wait! This means tangent has a problem. If the terminal side is vertical (like at \(\frac{\pi}{2}\) or \(\frac{3\pi}{2}\)), the \(x\)-coordinate is \(0\). You can't divide by zero, so \(\tan(\theta)\) is undefined at those points.

Key Takeaway:

\(\tan(\theta)\) is the slope of the terminal ray. It is undefined whenever the terminal ray is vertical (\(x=0\)).

6. Common Mistakes to Avoid

Don't worry if this seems tricky at first; even the best math students make these slips sometimes!

  • Mixing up Sine and Cosine: Always remember Cosine comes first alphabetically, just like \(x\) comes before \(y\). \((x, y) \rightarrow (\cos, \sin)\).
  • Calculator Mode: On Section I Part B and Section II Part A of the AP exam, your calculator must be in Radian Mode. If you use degree mode, your answers will be incorrect.
  • Forgetting the Radius: If you aren't on the Unit Circle (if \(r \neq 1\)), you must divide by \(r\). \(\sin(\theta)\) is not just \(y\); it is \(\frac{y}{r}\).

7. Summary Checklist

  • Can you draw an angle in standard position?
  • Do you know the definitions of \(\sin(\theta)\), \(\cos(\theta)\), and \(\tan(\theta)\) using \(x\), \(y\), and \(r\)?
  • Can you identify which functions are positive in each of the four quadrants?
  • Do you understand that \(\cos(\theta)\) is the \(x\)-coordinate and \(\sin(\theta)\) is the \(y\)-coordinate on the Unit Circle?
  • Do you recognize \(\tan(\theta)\) as the slope \(\frac{\sin \theta}{\cos \theta}\)?

Next Step: In Chapter 3.3, we will learn how to find the exact numerical values for specific angles like \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\). Keep this "coordinate" mindset ready!