【Math II】Trigonometric Functions: Master the Rhythm of Waves!

Hello! In Mathematics II, the topic that many students find intimidating because of its "endless list of formulas" is "Trigonometric Functions."
But don't worry. Trigonometric functions aren't just about random rote memorization; once you grasp the imagery of "rotation" and "waves," you will find it surprisingly clear and easy to understand.
Since this is a frequently appearing topic on the Common Test, let’s master it step by step together!

1. A New Way to Count Angles: "Circular Measure (Radians)"

Until now, you've used degrees like "30°" or "90°," but starting in Math II, the primary way of counting will be radians.
You might feel a bit lost at first, but the rules are actually very simple.

【Key Point】
Remember that \(180^{\circ} = \pi\) (pi)!
Once you have this down, you can figure out the rest through simple division:
・\(90^{\circ} = \frac{\pi}{2}\)
・\(60^{\circ} = \frac{\pi}{3}\)
・\(45^{\circ} = \frac{\pi}{4}\)
・\(30^{\circ} = \frac{\pi}{6}\)

Fun Fact:
Why do we do something so seemingly complicated? Because when you use radians, formulas for "arc length" and "area of a sector" become incredibly simple, appearing as \(l = r\theta\) and \(S = \frac{1}{2}r^2\theta\)!

2. General Angles and Definitions of Trigonometric Functions

In middle school and Math I, you thought about \(\sin, \cos, \tan\) using right-angled triangles. However, in Math II, we redefine them as "coordinates of points on a circle."
This allows us to handle angles greater than \(180^{\circ}\) and even negative angles.

Imagine a circle with a radius of \(1\) (this is called the unit circle).
For an angle \(\theta\), the coordinates of the point on the circle are:
・\(x\)-coordinate = \(\cos \theta\)
・\(y\)-coordinate = \(\sin \theta\)
・Slope = \(\tan \theta\)

Common Mistake:
Sometimes you might wonder, "Was \(\sin\) the \(x\) or the \(y\)?". When that happens, you can forcefully link it by remembering: "The \(y\)-coordinate is height, and sine is high (a high-flying image!)" to help it stick.

3. Interrelationships of Trigonometric Functions

There are three "super important" formulas that hold true for any angle.

① \(\tan \theta = \frac{\sin \theta}{\cos \theta\) (Tangent is sine over cosine)
② \(\sin^2 \theta + \cos^2 \theta = 1\) (Square them and add them to get 1!)
③ \(1 + \tan^2 \theta = \frac{1}{\cos^2 \theta}\)

Advice:
Instead of memorizing these just as "formulas," understand that they are simply applying the Pythagorean theorem to the unit circle mentioned earlier. You'll be much less likely to forget them that way.

4. Graphs of Trigonometric Functions: Capture the Wave Shape

Both \(\sin\) and \(\cos\) graphs have the shape of periodic waves.
・Period: The length it takes for one full set of the wave to complete. The base is \(2\pi\).
・Amplitude: The height of the wave. The base is \(1\) unit up and down.

Tips for reading graphs:
\(y = \sin \theta\) is a wave that starts at the origin (0, 0) and goes up.
\(y = \cos \theta\) is a wave that starts at the peak (0, 1) and goes down.
The shapes are the same; they are just shifted horizontally.

5. Addition Theorems: The Heart of Trigonometric Functions

This is the part that requires the most focus! The "Addition Theorem" is the parent of many other formulas.

【Formulas】
\(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\)
\(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\)

Classic Mnemonics:
(These rely on Japanese puns, but try to find your own rhythm!)
Just remember: Sine mixes the terms (\(\sin, \cos, \cos, \sin\)), while Cosine keeps the types together (\(\cos, \cos, -, \sin, \sin\)) and flips the sign.

It might feel difficult at first, but don't worry. If you chant them over and over, they will naturally start rolling off your tongue.

6. Double-Angle and Half-Angle Formulas

These are formulas derived from the Addition Theorem. It’s overwhelming to memorize everything, so try practicing deriving them from the Addition Theorem once—that way, you can reconstruct them yourself if you ever forget.

Especially important "Double-Angle Formulas":
\(\sin 2\theta = 2 \sin \theta \cos \theta\)
\(\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 1 - 2\sin^2 \theta = 2\cos^2 \theta - 1\)

Warning!
\(\cos 2\theta\) has three forms. The key is to choose the one that fits your problem: "Do I want an equation with only \(\sin\), or only \(\cos\)?".

7. Synthesis of Trigonometric Functions

This is a technique to combine two "separate waves," \(a \sin \theta + b \cos \theta\), into a single \(\sin\) expression.

Steps:
1. Plot the point \((a, b)\) using the coefficient of \(\sin\) (\(a\)) as the \(x\)-coordinate and the coefficient of \(\cos\) (\(b\)) as the \(y\)-coordinate.
2. Calculate the distance \(r = \sqrt{a^2 + b^2}\) from the origin to point \((a, b)\).
3. Find the angle \(\alpha\) formed with the positive direction of the \(x\)-axis.
4. You’ve got \(r \sin(\theta + \alpha)\)!

Why synthesize?
By consolidating into one expression, you can see the maximum and minimum values at a glance. This is used frequently in the latter half of the Common Test!

Key Takeaway

  • Radians: Think based on \(180^{\circ} = \pi\).
  • Definition: \(\sin\) is the \(y\)-coordinate, \(\cos\) is the \(x\)-coordinate.
  • Graphs: A wave that completes one cycle in \(2\pi\).
  • Addition Theorem: Master it perfectly with mnemonics.
  • Synthesis: Combine two waves into one to find the max and min.

Trigonometric functions become fun, like solving a puzzle, once you start seeing how the formulas connect.
Start by drawing the unit circle yourself and getting comfortable stating the basic angle values. I'm rooting for you!