【Math II】 Trigonometric Functions: Master Guide
Hello everyone! Welcome to the world of "Trigonometric Functions," one of the major peaks in Math II.
You might be thinking, "Aren't sin (sine) and cos (cosine) just those things from geometry we learned in junior high or Math I?" Well, in Math II, we're going to make angles much more dynamic and fluid. From the shape of waves and smartphone communication to digital music recording, the technology all around us is actually built on these functions!
They might look complicated at first, but if you break them down one by one, you'll definitely understand them. Let's conquer this together!
1. Expanding the Concept of Angles: General Angles and Radian Measure
What is a General Angle?
Until now, we only dealt with angles from 0° to 180° (or 360°), but in Math II, we also consider "angles that rotate around multiple times" and "angles that rotate in reverse."
Just imagine the hands of a clock. If it rotates backwards, it’s a negative angle; if it spins around twice, it’s 720°. We call these extended angles general angles.
A New Unit: "Radian Measure"
Starting in Math II, we primarily use a unit called "radian (rad)" instead of degrees (°).
We define 1 radian as the central angle formed when the arc length is equal to the radius.
Here is the most important point to remember:
\(180^\circ = \pi\) radians
【Tip: How to Convert】
Whenever you're unsure, just remember "\(\pi\) is 180°"!
・\(90^\circ = \frac{\pi}{2}\)
・\(60^\circ = \frac{\pi}{3}\)
・\(45^\circ = \frac{\pi}{4}\)
・\(30^\circ = \frac{\pi}{6}\)
*It’s helpful to think of it like slicing a pizza.
【Summary】
Angles go from "degrees" to "radians." Use \(180^\circ = \pi\) as your base for calculations!
2. Defining Trigonometric Functions: Think with the Unit Circle
We're going to rethink sin, cos, and tan—which we used to define as "side ratios" in triangles—on the "unit circle (a circle with radius 1)" on the coordinate plane. This is the true nature of trigonometric functions!
For a point \(P(x, y)\) on a unit circle, with angle \(\theta\):
・\(\cos \theta = x\)-coordinate
・\(\sin \theta = y\)-coordinate
・\(\tan \theta = \frac{y}{x}\) (the slope of the line)
【How to Remember】
Use the rhythm: "Horizontal (x) is cos, Vertical (y) is sin!"
Also, alphabetically, \(c (cos)\) comes before \(s (sin)\), just as \(x\) comes before \(y\) in coordinates, so they align perfectly as \(x = \cos, y = \sin\).
【Common Mistakes】
\(\tan \theta\) is undefined where \(\cos \theta = 0\) (i.e., \(90^\circ, 270^\circ \dots\)) because the denominator becomes zero. Be careful!
3. Relationships Between Trigonometric Functions
There are many formulas, but these three are the foundation. Think of it as a review of Math I.
① \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
② \(\sin^2 \theta + \cos^2 \theta = 1\)
③ \(1 + \tan^2 \theta = \frac{1}{\cos^2 \theta}\)
【Pro Tip】
Formula ② is just the Pythagorean theorem, \(a^2 + b^2 = c^2\), in disguise. Since the radius of the unit circle is 1, it becomes \(x^2 + y^2 = 1^2\), or \(\cos^2 \theta + \sin^2 \theta = 1\).
4. Graphs of Trigonometric Functions: The Essence of Waves
Trigonometric graphs are "periodic functions," meaning they repeat the same shape over and over.
\(y = \sin \theta\) and \(y = \cos \theta\)
・Both repeat their shape every \(2\pi\) (360°), so their period is \(2\pi\).
・Their values fluctuate between \(-1\) and \(1\).
・\(\sin\) is a wave starting from the origin \((0,0)\), while \(\cos\) starts at \((0,1)\).
\(y = \tan \theta\)
・The period is \(\pi\) (This is a common trap!).
・It has a unique shape that stretches up and down. Its characteristic is having asymptotes (lines that the graph approaches but never touches).
【Strategy Tip】
If a problem asks you to graph \(y = \sin 2\theta\), think of it as "being horizontally compressed by a factor of 2." Since the angle advances at twice the speed, the period becomes half (\(\pi\)).
5. The Most Important Part: Addition Theorems
This is the most crucial formula in Math II trigonometry. Once you master this, you can derive the double-angle and half-angle formulas yourself!
Addition Theorem for \(\sin\)
\(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\)
Addition Theorem for \(\cos\)
\(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\)
*Note: Be careful, as the sign for \(\cos\) is the opposite (it’s a minus when the formula has a plus)!
Addition Theorem for \(\tan\)
\(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\)
【Summary】
Recite these like a magic spell until you memorize them. With these, you can calculate values like \(75^\circ\) (\(45^\circ + 30^\circ\)) with ease!
6. Applications: Double-Angle, Half-Angle, and Synthesis
These are convenient formulas derived from the addition theorems. You might feel overwhelmed at first, but you'll get used to the patterns.
Double-Angle Formulas
・\(\sin 2\alpha = 2 \sin \alpha \cos \alpha\)
・\(\cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha = 1 - 2 \sin^2 \alpha = 2 \cos^2 \alpha - 1\)
*It’s super important to remember that \(\cos 2\alpha\) can transform into three different forms!
Synthesis of Trigonometric Functions
This is a technique to combine the expression \(a \sin \theta + b \cos \theta\) into a single \(\sin\) function.
\(a \sin \theta + b \cos \theta = \sqrt{a^2 + b^2} \sin(\theta + \alpha)\)
This is incredibly useful when finding the height or maximum value of a wave.
【Steps for Synthesis】
1. Plot point \((a, b)\) on the coordinate plane.
2. Calculate the distance \(\sqrt{a^2+b^2}\) from the origin (this is the new amplitude).
3. Determine the angle \(\alpha\) relative to the positive \(x\)-axis.
Final Advice: Study Tips
Trigonometry can look daunting with all these formulas, but remember: "All formulas are connected to the addition theorem." Instead of rote memorization, try deriving the formulas on a blank sheet of paper—you'll be surprised at how well they stick!
It might feel difficult at first, but don't worry. As you draw the unit circle over and over and visualize the waves, there will definitely be an "Aha!" moment. Take it one step at a time!