Trig. functions

Welcome to the World of Waves: Trig Functions!

Welcome to one of the most visual and "loopy" chapters in Mathematics B (MEI)! In this section, we move beyond just finding missing sides in triangles (SOH CAH TOA) and start looking at trigonometry as functions that repeat forever. These functions are the "heartbeat" of math—they describe everything from the tides of the ocean to the sound waves in your headphones. Don't worry if this seems tricky at first; once you see the patterns, it all starts to click!

1. The Unit Circle: Trigonometry Beyond Triangles

When you first learned trig, you used right-angled triangles. But what happens if the angle is 150°, or even -45°? You can't draw a triangle with those! To solve this, we use the Unit Circle—a circle with a radius of exactly 1, centered at (0,0).

How it works:

Imagine a point moving around this circle. The angle \(\theta\) starts from the positive x-axis and turns anti-clockwise. The coordinates of that point \((x, y)\) actually define our trig functions:

• Cosine is the x-coordinate: \(\cos \theta = x\)
• Sine is the y-coordinate: \(\sin \theta = y\)
• Tangent is the slope: \(\tan \theta = \frac{y}{x}\)

Quick Tip: Think "C" for Cosine and "X" (they are both near the end of the alphabet). Think "S" for Sine and "Y" (they are both... well, not X!).

The CAST Diagram

Since the point moves through four quadrants, the values of sin, cos, and tan change from positive to negative. We use the CAST diagram to remember which is positive where:

• Quadrant 1 (0-90°): All are positive.
• Quadrant 2 (90-180°): Sine is positive.
• Quadrant 3 (180-270°): Tan is positive.
• Quadrant 4 (270-360°): Cos is positive.

Memory Aid: "All Stations To Crewe" or "Add Sugar To Coffee".

Key Takeaway: The Unit Circle allows us to define trig values for any angle, not just the ones inside a triangle.

2. The "Hall of Fame": Exact Values

The MEI syllabus requires you to know certain trig values by heart. You can always use a calculator, but knowing these will save you massive amounts of time in "non-calculator" style questions.

Degrees and Radians

In A-Level, we use Radians alongside degrees. Remember: \(180^\circ = \pi\) radians.

The Values You Need:

• 0° (0 rad): \(\sin(0)=0\), \(\cos(0)=1\), \(\tan(0)=0\)
• 30° (\(\frac{\pi}{6}\) rad): \(\sin(30)=\frac{1}{2}\), \(\cos(30)=\frac{\sqrt{3}}{2}\), \(\tan(30)=\frac{1}{\sqrt{3}}\)
• 45° (\(\frac{\pi}{4}\) rad): \(\sin(45)=\frac{1}{\sqrt{2}}\), \(\cos(45)=\frac{1}{\sqrt{2}}\), \(\tan(45)=1\)
• 60° (\(\frac{\pi}{3}\) rad): \(\sin(60)=\frac{\sqrt{3}}{2}\), \(\cos(60)=\frac{1}{2}\), \(\tan(60)=\sqrt{3}\)
• 90° (\(\frac{\pi}{2}\) rad): \(\sin(90)=1\), \(\cos(90)=0\), \(\tan(90)\) is undefined

Did you know? You can find the 30° and 60° values just by drawing an equilateral triangle with side length 2 and cutting it in half!

Key Takeaway: Memorizing these values is like learning your times tables—it makes the harder stuff much easier.

3. Trig Graphs: The Visual Waves

When you plot trig functions on a graph, they create beautiful, repeating patterns called periodic functions.

The Sine Graph: \(y = \sin \theta\)

• Starts at (0,0).
• Waves between 1 and -1 (this is called the Amplitude).
• Repeats every 360° (this is the Period).
• Has rotational symmetry about the origin.

The Cosine Graph: \(y = \cos \theta\)

• Starts at the top (0,1).
• Looks exactly like the sine graph but shifted 90° to the left.
• Is symmetrical across the y-axis (it's an even function).

The Tangent Graph: \(y = \tan \theta\)

• Looks like a series of "S" shapes.
• Has vertical asymptotes (lines the graph never touches) at 90°, 270°, etc.
• Has a shorter period: it repeats every 180°.

Common Mistake: Forgetting that \(\tan \theta\) repeats twice as often as \(\sin \theta\) and \(\cos \theta\)!

Key Takeaway: Sine starts at the center; Cosine starts at the top; Tan has "walls" (asymptotes) it can't cross.

4. Transforming Trig Graphs

Just like with algebraic functions, you can stretch, squish, and move trig graphs. This is a common exam topic!

Step-by-Step Guide to Transformations:

1. \(y = a\sin(x)\): This is a vertical stretch. It changes the amplitude. If \(a=3\), the wave goes up to 3 and down to -3.
2. \(y = \sin(bx)\): This is a horizontal stretch. It changes the period. The new period is \(\frac{360}{b}\). So, \(\sin(2x)\) repeats every 180° (it's twice as fast!).
3. \(y = \sin(x) + d\): This is a vertical translation. It shifts the whole wave up or down.
4. \(y = \sin(x + c)\): This is a horizontal translation. It shifts the wave left (if \(c\) is positive) or right (if \(c\) is negative).

Analogy: Imagine the trig graph is a spring. Changing \(a\) pulls it taller; changing \(b\) squashes it together; changing \(c\) and \(d\) just moves the whole spring around the room.

Key Takeaway: Numbers outside the function affect the y-axis (vertical); numbers inside the function affect the x-axis (horizontal) and usually do the opposite of what you'd expect!

5. Inverse Trig Functions: Going Backwards

If \(\sin(30) = 0.5\), then the inverse tells us that the angle for 0.5 is 30°. We write these as \(\arcsin\), \(\arccos\), and \(\arctan\) (or \(\sin^{-1}, \cos^{-1}, \tan^{-1}\)).

The Domain and Range Trap

Because trig graphs repeat forever, your calculator can only give you one answer (the Principal Value). We restrict the range so the inverse functions actually work:

• \(\arcsin(x)\): Output is between -90° and 90° (\(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\)).
• \(\arccos(x)\): Output is between 0° and 180° (\(0\) to \(\pi\)).
• \(\arctan(x)\): Output is between -90° and 90° (\(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\)).

Quick Review Box:
Function: \(\sin \theta\). Inverse: \(\arcsin(x)\)
Graph: The inverse graph is a reflection of the original graph in the line \(y = x\).

Key Takeaway: Inverse functions find the angle, but they only give you the "main" answer. Use the graph's symmetry to find other solutions in a given range!

Summary Checklist

Before you move on to Trig Identities, make sure you can:

• Draw the Unit Circle and explain why \(\sin \theta = y\) and \(\cos \theta = x\).
• Recite the Exact Values for 0, 30, 45, 60, and 90 degrees.
• Sketch the Sine, Cosine, and Tan graphs including their periods and asymptotes.
• Apply transformations like \(y = 2\cos(x) + 1\) to a graph.
• Identify the Principal Values of inverse trig functions.