Special Trig Values

Statement

Certain trigonometric ratios take especially simple values when the angles are 30°, 45°, or 60°. These “special trig values” are often required in GCSE exams, both for non-calculator questions and to simplify exact answers.

\[ \sin 45^\circ = \cos 45^\circ = \tfrac{\sqrt{2}}{2}, \quad \sin 30^\circ = \tfrac{1}{2}, \quad \cos 30^\circ = \tfrac{\sqrt{3}}{2}, \quad \sin 60^\circ = \tfrac{\sqrt{3}}{2}, \quad \cos 60^\circ = \tfrac{1}{2} \]

Why it’s true

• These values come from geometry, not memorisation alone. By constructing special right-angled triangles, we can derive them.
• A 45°–45°–90° triangle (isosceles right triangle) shows that both legs are equal, giving sine and cosine of 45° equal to \( \frac{\sqrt{2}}{2} \).
• An equilateral triangle cut in half gives 30°–60°–90° triangles. Using Pythagoras, we derive the ratios for sine and cosine of 30° and 60°.

Recipe (how to use it)

1. Identify if the question uses one of the angles 30°, 45°, or 60°.
2. Recall or reconstruct the special triangle to write down the exact value.
3. Use the value in calculations, simplifying surds when necessary.

Spotting it

Look for exam questions with exact angles such as 30°, 45°, or 60°. Calculator questions usually give decimals, but “exact value” questions want you to use these special ratios.

Common pairings

• Exact trig values appear in Pythagoras-style or surd simplification questions.
• They often pair with trigonometric identities, such as \(\sin^2\theta + \cos^2\theta = 1\).
• Geometry questions involving equilateral or isosceles right-angled triangles.

Mini examples

1. Given: Find \(\sin 30^\circ\). Answer: \(\tfrac{1}{2}\).
2. Given: Simplify \(\cos 60^\circ \times 4\). Answer: \(2\).

Pitfalls

• Confusing sine and cosine values for 30° and 60°.
• Forgetting that the values must remain in surd form (e.g. writing 0.707 instead of \(\tfrac{\sqrt{2}}{2}\)).
• Using a calculator when the question says “exact value”.
• Dropping the denominator when simplifying surds.

Exam strategy

• Memorise the triangles or the values; both methods are valid.
• Check if the exam requires an exact answer. Never approximate unless asked.
• Write fractions clearly, with surds simplified when possible.

Summary

The special trig values for 30°, 45°, and 60° give exact surd results instead of decimals. They are crucial in non-calculator exams, algebraic trigonometry, and proofs. Remembering them saves time and prevents mistakes when working with exact trigonometric expressions.