Introduction: Why Exact Values Matter

Hi there! Have you ever noticed that your calculator gives you long, messy decimals like 0.866025... when you type in \(\sin(60^\circ)\)? In A Level Mathematics, we prefer to be precise. Instead of rounding those decimals, we use Exact Values—expressions involving surds and fractions that are 100% accurate.

Learning these values is a superpower for your OCR H240 exams. It makes solving complex trigonometric equations much faster and helps you avoid rounding errors that could cost you marks. Don't worry if it seems like a lot to memorize at first; we have some great tricks to make it easy!


1. The Prerequisite: Special Triangles

The easiest way to understand where these values come from is by looking at two "special" triangles. If you can sketch these, you never have to "memorize" the table!

The Isosceles Right-Angled Triangle (\(45^\circ\) or \(\frac{\pi}{4}\) radians)

Imagine a right-angled triangle where the two shorter sides are both 1 unit long. Using Pythagoras, the longest side (hypotenuse) must be \(\sqrt{2}\).

From this triangle, we can see:
\(\sin(45^\circ) = \frac{1}{\sqrt{2}}\) (which is the same as \(\frac{\sqrt{2}}{2}\))
\(\cos(45^\circ) = \frac{1}{\sqrt{2}}\)
\(\tan(45^\circ) = \frac{1}{1} = 1\)

The Equilateral Triangle (\(30^\circ\) and \(60^\circ\))

Imagine an equilateral triangle where every side is 2 units long. If we chop it in half down the middle, we get a right-angled triangle with a base of 1, a hypotenuse of 2, and a height of \(\sqrt{3}\).

Using this "half-triangle":
For \(30^\circ\) (\(\frac{\pi}{6}\)): \(\sin(30^\circ) = \frac{1}{2}\), \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\)
For \(60^\circ\) (\(\frac{\pi}{3}\)): \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\), \(\cos(60^\circ) = \frac{1}{2}\), \(\tan(60^\circ) = \sqrt{3}\)

Key Takeaway: If you forget a value in the exam, quickly sketch a 1-1-\(\sqrt{2}\) triangle or a 1-\(\sqrt{3}\)-2 triangle to find it!


2. The Exact Values Table

Here are the core values you need to know for Stage 1 and Stage 2 of the OCR syllabus. These are the "building blocks" for the rest of trigonometry.

Sine, Cosine, and Tangent Values

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

Did you know? The word "sine" comes from the Latin word sinus, which means "bay" or "curve." The values of sine and cosine are actually just coordinates on a circle with a radius of 1!


3. Memory Aids: The Finger Trick

Struggling to remember the table? Try the Left Hand Rule:

  1. Hold out your left hand, palm facing you.
  2. Your thumb is \(90^\circ\), index is \(60^\circ\), middle is \(45^\circ\), ring is \(30^\circ\), and pinky is \(0^\circ\).
  3. To find a value, fold down that finger.
  4. Sine: \(\frac{\sqrt{\text{fingers below}}}{2}\)
  5. Cosine: \(\frac{\sqrt{\text{fingers above}}}{2}\)

Example: Fold your ring finger (\(30^\circ\)). There is 1 finger below (pinky). So \(\sin(30^\circ) = \frac{\sqrt{1}}{2} = \frac{1}{2}\). Magic!


4. Moving Beyond \(90^\circ\): Multiples and Symmetry

The syllabus requires you to know values for \(180^\circ\) (\(\pi\)) and multiples of the basic angles (like \(210^\circ\) or \(\frac{3\pi}{4}\)). We use the CAST Diagram to figure these out.

The CAST Diagram Analogy

Think of the four quadrants like four different clubs. Every club has a "House Rule" about who is allowed to be Positive:

  • Quadrant 1 (\(0\) to \(90^\circ\)): All are positive.
  • Quadrant 2 (\(90\) to \(180^\circ\)): Only Sine is positive.
  • Quadrant 3 (\(180\) to \(270^\circ\)): Only Tangent is positive.
  • Quadrant 4 (\(270\) to \(360^\circ\)): Only Cosine is positive.

Mnemonic: All Science Teachers Care (or Add Sugar To Coffee).

How to find a multiple (Step-by-Step):

Example: Find the exact value of \(\cos(210^\circ)\).

  1. Find the Quadrant: \(210^\circ\) is in the 3rd Quadrant (between \(180^\circ\) and \(270^\circ\)).
  2. Find the Reference Angle: How far is \(210^\circ\) from the horizontal (\(180^\circ\))? \(210 - 180 = 30^\circ\).
  3. Find the Value: We know \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\).
  4. Check the Sign: In the 3rd Quadrant (T), only Tan is positive. So Cosine must be negative.
  5. Final Answer: \(\cos(210^\circ) = -\frac{\sqrt{3}}{2}\).

Quick Review Box:
- \(\sin(180^\circ) = 0\)
- \(\cos(180^\circ) = -1\)
- \(\tan(180^\circ) = 0\)


5. Common Mistakes to Avoid

Even the best students trip up on these! Keep an eye out for:

  • Calculator Mode: Always check if your calculator is in Degrees or Radians. If the question has a \(\pi\), you should almost always be in Radians!
  • The Undefined Tangent: \(\tan(90^\circ)\) and \(\tan(270^\circ)\) do not have values. If you see these in an equation, look for a vertical asymptote on the graph.
  • Confusing Sine and Cosine: Remember that as the angle increases from \(0\) to \(90^\circ\), Sine goes UP (0 to 1) and Cosine goes DOWN (1 to 0).

Summary Checklist

Key Takeaways:

  • Can you sketch the \(30/60\) and \(45\) degree triangles from memory?
  • Do you know the exact values for \(\sin\), \(\cos\), and \(\tan\) at \(0, 30, 45, 60, 90\)?
  • Can you convert these to Radians (\(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\))?
  • Can you use the CAST diagram to find values like \(\sin(120^\circ)\) or \(\cos(\pi)\)?

Keep practicing! Trigonometry is like a language—the more you speak it, the more natural these values will feel.