Exact Values Table: Sin, Cos, Tan for All Special Angles

The complete exact values table for trigonometry, covering sin, cos, and tan for every special angle from 0° to 360°. Bookmark this page as your go-to reference for exams and problem sets.

If you have ever searched for an exact values table trigonometry reference, you have come to the right place. Below you will find two comprehensive tables: one for the five core special angles, and one extended table covering all quadrants using reference angles. Every value is given in exact form, no decimals, no rounding, just the precise algebraic expressions your teacher expects in your working.

What Are Exact Values in Trigonometry?

Exact values are the precise results of trigonometric functions at specific angles, expressed using integers, fractions, and square roots rather than decimal approximations. For example, sin 30° is exactly 12, not 0.5. While both are numerically equivalent in that case, for most angles the decimal is an approximation whereas the exact form is perfectly precise.

Exact values matter because they allow you to work through multi-step problems without accumulating rounding errors. They are essential for solving trigonometric equations, proving identities, and evaluating definite integrals. In short, if you are studying maths at any level beyond basic geometry, you need these values at your fingertips. For a deeper discussion, see our article on why memorising exact values is essential.

The Core Exact Values Table (0° to 90°)

These five special angles form the foundation of all trigonometric exact values. Every other angle in the unit circle maps back to one of these through reference angles and sign changes.

Table 1 — The five core special angles and their exact trigonometric values

Notice the elegant symmetry: the sine values from 0° to 90° are 0, 12, 1√2, √32, 1, while the cosine values are the exact reverse: 1, √32, 1√2, 12, 0. This is because sin θ = cos(90° − θ), a complementary relationship that cuts your memorisation effort in half.

Understanding Reference Angles

A reference angle is the acute angle formed between the terminal side of your angle and the x-axis. It is always between 0° and 90°, which means it always corresponds to one of the five special angles in the core table above.

To find the exact value of a trigonometric function at any angle:

1. Find the reference angle — Determine how far the angle is from the nearest x-axis direction (0° or 180°).
2. Look up the core value — Use the table above for that reference angle.
3. Apply the correct sign — Use the CAST rule to decide whether the result is positive or negative in that quadrant.

The CAST Rule

The CAST diagram tells you which trigonometric functions are positive in each quadrant. The name comes from reading anticlockwise starting from the fourth quadrant:

For example, to find sin 150°: the reference angle is 180° − 150° = 30°. We are in Quadrant II where sine is positive, so sin 150° = +sin 30° = 12. Similarly, cos 150° = −cos 30° = −√32 because cosine is negative in Quadrant II.

Extended Exact Values Table (0° to 360°)

Using reference angles and the CAST rule, the five core values expand into the full table below. This is the complete exact values table for all standard angles in trigonometry.

Table 2 — Extended exact values for all standard angles from 0° to 360°

Worked Example: Finding sin 225°

Let us walk through a full example step by step.

1. Identify the quadrant. 225° is between 180° and 270°, so it falls in Quadrant III.
2. Find the reference angle. 225° − 180° = 45°.
3. Look up the core value. From Table 1, sin 45° = 1√2.
4. Apply the sign. In Quadrant III, only tangent is positive (the T in CAST). Sine is negative here, so sin 225° = −1√2.

That same three-step process works for every single angle in the extended table. Once you know the five core values and the CAST rule, you never need to look up a table again.

How to Remember Exact Values

There are several popular memory techniques that make the core table easy to internalise:

• The counting pattern. For sin, the numerators follow √0, √1, √2, √3, √4 over a denominator of 2. Since √0 = 0 and √4 = 2, this simplifies neatly to the values in the table.
• The hand trick. Hold up your left hand with fingers spread. Each finger from thumb to pinkie represents 0°, 30°, 45°, 60°, 90°. Fold down the finger for your angle, count remaining fingers on each side, and apply a simple formula.
• Complementary pairs. Since sin θ = cos(90° − θ), you only need to memorise either the sine or cosine row, not both. The other is simply the reverse.
• Practice, practice, practice. Nothing beats active recall. Testing yourself repeatedly is the fastest path to long-term retention.

For more tips and a deeper discussion on building lasting memory, read our full guide: Why We Need to Memorise Exact Values in Mathematics.

Why This Table Matters for Exams

In many exam systems, particularly the VCE, HSC, and IB, technology-free sections require you to evaluate trigonometric expressions without a calculator. If you cannot recall exact values from memory, you either waste time re-deriving them or risk making errors under pressure.

Having this exact values table internalised means you can instantly recognise expressions like sin(5π6) = 12 and move on to the harder parts of the question. It transforms trigonometry from a stumbling block into a strength.

Ready to lock these values into memory? Our interactive quiz tests you on every angle and function until they are second nature.