Introduction: The Rhythm of Mathematics

Welcome! In this section, we are diving into the Graphs of the Basic Trigonometric Functions. You’ve likely used \(\sin\), \(\cos\), and \(\tan\) to find sides and angles in triangles, but now we are going to look at them as functions that go on forever.

Why does this matter? Because trigonometry is the language of waves. From the sound coming out of your headphones to the way light travels and how tides move, everything that repeats in a cycle can be modeled using these graphs. Don't worry if you find the shapes a bit strange at first; by the end of these notes, you’ll be able to sketch them with confidence!

1. Prerequisite: Degrees and Radians

Before we draw, remember that in A Level Mathematics, we use two units for angles:
1. Degrees (\(^\circ\)): The classic \(0^\circ\) to \(360^\circ\).
2. Radians (rad): The "mathematical" unit where a full circle is \(2\pi\).
Key conversion: \(180^\circ = \pi\) radians.

2. The Sine Function: \(y = \sin x\)

The sine graph is often called a sine wave. It represents a smooth, periodic oscillation.

Key Features:

  • Shape: A continuous wave that starts at the origin \((0,0)\).
  • Period: The wave repeats every \(360^\circ\) (or \(2\pi\) radians).
  • Amplitude: The graph goes as high as \(1\) and as low as \(-1\).
  • Symmetry: It has rotational symmetry about the origin (it is an odd function).

Step-by-Step Sketching (\(0^\circ\) to \(360^\circ\)):

1. Start at \((0,0)\).
2. Go up to the peak at \((90^\circ, 1)\).
3. Come back down to the x-axis at \((180^\circ, 0)\).
4. Continue down to the "valley" at \((270^\circ, -1)\).
5. Finish the cycle back at \((360^\circ, 0)\).

Quick Review Box:
Domain: All real numbers (\(x \in \mathbb{R}\))
Range: \(-1 \leq y \leq 1\)

3. The Cosine Function: \(y = \cos x\)

The cosine graph looks almost exactly like the sine graph, but it is shifted to the left by \(90^\circ\).

Key Features:

  • Shape: A wave that starts at its maximum value \((0,1)\).
  • Period: Just like sine, it repeats every \(360^\circ\) (or \(2\pi\)).
  • Amplitude: Stays between \(1\) and \(-1\).
  • Symmetry: It is symmetrical across the y-axis (it is an even function).

Memory Aid: The "Cosy Cup"

A simple trick to remember the shape of \(\cos x\) between \(0^\circ\) and \(360^\circ\) is that it looks like a Cup (starting high, dipping down, and ending high).

Did you know?
The word "cosine" comes from "complementary sine." This is because \(\cos x = \sin(90^\circ - x)\). They are essentially the same wave, just "out of phase" with each other!

Key Takeaway:

The sine and cosine graphs are "waves" that never go above \(1\) or below \(-1\). They repeat every \(360^\circ\).

4. The Tangent Function: \(y = \tan x\)

The tangent graph is the "rebel" of the group. It doesn't look like a wave at all!

Key Features:

  • Shape: A series of repeating "curves" separated by vertical gaps.
  • Period: It repeats much faster, every \(180^\circ\) (or \(\pi\) radians).
  • Asymptotes: These are the vertical gaps where the function is undefined. They occur at \(90^\circ, 270^\circ, -90^\circ\), etc. The graph gets closer and closer to these lines but never touches them.
  • Range: Unlike sine and cosine, tangent goes from \(-\infty\) to \(+\infty\).

Real-World Analogy:

Think of \(\tan x\) as the slope of a hill. At \(0^\circ\), the ground is flat (slope = 0). As you get closer to \(90^\circ\), the hill gets steeper and steeper until it's a vertical wall—impossible to climb! That's why \(\tan 90^\circ\) is undefined.

Common Mistake to Avoid:
Students often forget that the period of \(\tan x\) is \(180^\circ\), not \(360^\circ\). Always double-check your x-axis scale when sketching \(\tan\)!

5. Exact Values You Must Know

For OCR H240, you are expected to know the exact values for specific angles without a calculator. These are often used in non-calculator papers.

Common Angles Table:

\(x = 0^\circ (0)\): \(\sin x = 0\), \(\cos x = 1\), \(\tan x = 0\)
\(x = 30^\circ (\frac{\pi}{6})\): \(\sin x = \frac{1}{2}\), \(\cos x = \frac{\sqrt{3}}{2}\), \(\tan x = \frac{1}{\sqrt{3}}\)
\(x = 45^\circ (\frac{\pi}{4})\): \(\sin x = \frac{1}{\sqrt{2}}\), \(\cos x = \frac{1}{\sqrt{2}}\), \(\tan x = 1\)
\(x = 60^\circ (\frac{\pi}{3})\): \(\sin x = \frac{\sqrt{3}}{2}\), \(\cos x = \frac{1}{2}\), \(\tan x = \sqrt{3}\)
\(x = 90^\circ (\frac{\pi}{2})\): \(\sin x = 1\), \(\cos x = 0\), \(\tan x = \text{Undefined}\)

The "Hand Trick" Memory Aid:

Hold up your left hand, palm facing you. Let your pinky be \(0^\circ\) and your thumb be \(90^\circ\). To find \(\sin\) or \(\cos\) of an angle, fold down that finger (\(30^\circ\) is your ring finger, etc.).
\(\sin = \frac{\sqrt{\text{fingers below}}}{2}\)
\(\cos = \frac{\sqrt{\text{fingers above}}}{2}\)

6. Symmetry and Periodicity

Because these graphs repeat, you can find the value of an angle like \(210^\circ\) by using the symmetry of the graph.

  • Sine Symmetry: \(\sin(180^\circ - \theta) = \sin \theta\). Example: \(\sin 150^\circ = \sin 30^\circ = 0.5\).
  • Cosine Symmetry: \(\cos(360^\circ - \theta) = \cos \theta\). Example: \(\cos 300^\circ = \cos 60^\circ = 0.5\).
  • Periodicity: You can add or subtract the period to any angle. \(\sin(x + 360^\circ) = \sin x\).
Key Takeaway:

If you know the values and shape of the graphs between \(0^\circ\) and \(90^\circ\), you can use symmetry to find the value of any angle in the world!

Final Quick Review

1. \(\sin x\): Starts at \(0\), max at \(90^\circ\), period \(360^\circ\).
2. \(\cos x\): Starts at \(1\), zero at \(90^\circ\), period \(360^\circ\).
3. \(\tan x\): Vertical asymptotes at \(90^\circ, 270^\circ\), period \(180^\circ\).
4. Exact Values: Memorize the \(30^\circ/45^\circ/60^\circ\) values—they are your best friends in exams!