Trig Exact Value Chart: Sin, Cos and Tan Table You Can Rebuild in 20 Seconds

The Trig Exact Value Chart

This is the complete trigonometry table for the five special angles. These are the only exact trig values you are expected to recall in a non‑calculator paper.

You will see this same exact trig values table called several different things — a sin cos tan table, a trig table, a trigonometry table, a trig values table or an exact value trig chart. They all mean the chart above, so if you arrived here searching for one of those, you are in the right place.

How to Read the Exact Values Table

Pick your angle in the left column, then read across to the ratio you need. Two worked reads:

• sin 60° — find the 60° row, move to the sin column: √3/2. As a decimal that is 0.866, but in a non‑calculator paper you leave it as √3/2.
• tan 30° — find the 30° row, move to the tan column: √3/3. You will also see this written as 1/√3. They are the same number; see the note below.

Why tan 30° is sometimes 1/√3 and sometimes √3/3

Both are correct and both equal 0.5773… They differ only by rationalising the denominator — multiplying top and bottom by √3 to clear the surd out of the bottom of the fraction:

Most exam boards accept either form unless the question says “give your answer with a rational denominator”. If in doubt, write √3/3 — it is never marked wrong.

Where the Exact Trig Values Come From: the Exact Value Triangles

This is the part most revision pages skip, and it is the part that makes the chart impossible to forget. Every value in the table comes from just two triangles. If you can draw these two shapes, you never need the table again.

Triangle 1: the 45‑45‑90 triangle

Start with a square of side 1 and cut it along the diagonal. You get a right‑angled triangle with two legs of 1 and two equal angles, so each must be 45°. Pythagoras gives the hypotenuse:

Now just read the ratios off the shape, using SOH‑CAH‑TOA with 45° as the angle:

That is the whole 45° row, derived in three lines. Notice why sin 45° and cos 45° are equal: the two legs are the same length, so “opposite” and “adjacent” are the same number.

Triangle 2: the 30‑60‑90 triangle

Start with an equilateral triangle of side 2. All three angles are 60°. Drop a perpendicular from the apex to the base: it cuts the base exactly in half (giving 1 and 1) and cuts the 60° apex angle exactly in half (giving 30°). Pythagoras gives the height:

This single triangle gives you two rows of the chart, because it contains both a 30° angle and a 60° angle. The only thing that changes is which side counts as “opposite”.

Looking at the 30° angle at the top. The side opposite it is the short one (1). The side next to it is the long one (√3). The hypotenuse is 2.

Now looking at the 60° angle at the bottom. Opposite and adjacent swap over: the side opposite 60° is the long one (√3), and the side next to it is the short one (1).

What about 0° and 90°?

These two are not from a triangle — you cannot have a triangle with a 0° angle. Instead, picture a right‑angled triangle and imagine squashing the angle down towards 0° or opening it up towards 90°.

• As the angle shrinks to 0° the opposite side shrinks to nothing while the adjacent side becomes the whole hypotenuse. So sin 0° = 0/1 = 0 and cos 0° = 1/1 = 1. Then tan 0° = 0 ÷ 1 = 0.
• As the angle opens to 90° the opposite side becomes the whole hypotenuse while the adjacent side shrinks to nothing. So sin 90° = 1/1 = 1 and cos 90° = 0/1 = 0. Then tan 90° = 1 ÷ 0, which has no answer — see the section on why tan 90° is undefined.

The Exact Trig Values Hand Trick

If drawing triangles feels slow under exam pressure, there is a faster method that uses nothing but your left hand. It gives you sin and cos for all five angles in a couple of seconds, and it works because of the √n ÷ 2 pattern already noted at the top of this page.

Step 1: label your fingers

Hold up your left hand, palm towards you. Starting at the thumb and moving across to the little finger, the five fingers stand for the five angles: 0°, 30°, 45°, 60°, 90°.

Step 2: fold down the finger for your angle

Say you want sin 60°. Fold down the ring finger, because that is the 60° finger. Now count the fingers left standing on each side of the fold:

• sin = √(number of fingers below the fold) ÷ 2 — below meaning on the thumb side.
• cos = √(number of fingers above the fold) ÷ 2 — above meaning on the little‑finger side.

In the example above, folding the 60° finger leaves three fingers below it and one above:

Both match the chart exactly. Here is every angle worked through the same way, so you can check the trick against the table:

The trick is not magic. It works because the number of fingers below the fold counts up 0, 1, 2, 3, 4 exactly as the angle grows — which is the same √n ÷ 2 pattern the chart follows. The hand is just a way of counting it without writing anything down.

Exact Trig Values in Radians

If you are doing IB Maths, A‑level or anything beyond GCSE, you will meet these same exact values written in radians rather than degrees. Nothing about the values changes — only the label on the angle.

A radian is just a different unit for measuring turn, in the same way that kilometres and miles are different units for measuring distance. The one fact that connects them:

Everything else follows from that single line. To convert in either direction:

So 60° becomes 60 × π/180 = π/3, which is the value already shown in the radians column of the chart. Working the other way, π/4 becomes (π/4) × (180/π) = 45°.

Why IB and A‑level use radians

It is not to make life harder. Radians make the calculus of trigonometry work cleanly — the derivative of sin x is cos x only when x is in radians. In degrees you would be stuck carrying an awkward factor of π/180 through every single line. Arc length and sector area formulas are also far simpler in radians.

If you are heading into IB Maths, it is worth getting comfortable reading the radian column of the chart as fluently as the degrees column, because past the first year you will rarely see degrees again.

Trigonometric Values Beyond 90°: the Unit Circle

The five‑angle chart covers 0° to 90°. But exam questions ask for things like sin 150° or cos 240° too. You do not need to learn a second, longer table — every one of those values is a value you already know, with a sign attached.

The idea: reference angles

Picture a circle of radius 1 with its centre at the origin. Measure your angle anticlockwise from the positive x‑axis, and mark the point where that angle lands on the circle. The coordinates of that point are exactly (cos θ, sin θ).

Angles past 90° land in the other three quarters of the circle, but they are always the same distance from the horizontal axis as one of your five known angles. That known angle is called the reference angle. So:

• 150° is 180° − 30°, so its reference angle is 30°. The size of every ratio matches 30°; only the signs may differ.
• 240° is 180° + 60°, so its reference angle is 60°.
• 315° is 360° − 45°, so its reference angle is 45°.

Find the reference angle, look up its value in the five‑angle chart, then decide the sign using the rule in the next section. That is the entire method.

The full exact trig values table, 0° to 360°

Here is every exact trigonometric value you will be asked for in a full turn. Notice that no new numbers appear — it is the same short list of values from the five‑angle chart, repeating with different signs.

Read down the sin column and you will see the values of trigonometric ratios repeat: 0, ½, √2/2, √3/2, 1 climb and fall and climb again. There are only five distinct sizes in the whole table. Learn those five and the signs, and you have all seventeen rows.

Which Signs Are Positive: the CAST Rule

Once you have the reference angle, you need one more thing: the sign. Whether sin, cos or tan comes out positive or negative depends only on which quarter of the circle your angle lands in, and the pattern never changes.

The memory aid is the word CAST, read starting from the bottom‑right quadrant and moving anticlockwise.

Putting it together: a worked example

Why tan 90° Is Undefined

Every other cell in the chart holds a number. The tan 90° cell says “undefined”, and it is worth understanding why rather than just memorising the word.

Since tan is sin divided by cos:

Dividing by zero has no answer, so tan 90° simply does not exist. The same happens at 270°, where cos is also zero.

There is a nice way to picture it. On the unit circle, tan θ is the gradient of the line from the origin out to your point. As the angle climbs towards 90° that line gets steeper and steeper, and a perfectly vertical line has no gradient at all — it rises without ever moving sideways.

Common Mistakes With the Trigonometry Table

These are the slips that cost marks most often. None of them are about not knowing the values — they are about using them carelessly.

Practice: Exact Trig Values Questions

Work each one out on paper first, then tap to check. They get harder as you go. Every answer below has been checked symbolically, so if your answer disagrees, look again at your working.

Trig Exact Value Chart FAQs

Putting the Chart to Work

The exact values are rarely the whole question. They turn up inside something larger — solving a trig equation, simplifying an expression, finding an area, working through a problem in IB Maths. What examiners are really checking is whether you can produce the right value quickly and attach the right sign, without reaching for a calculator you are not allowed.

So the useful test is not “can I recite the table”. It is this: given any angle between 0° and 360°, in degrees or radians, can you write down its sin, cos and tan exactly, in under ten seconds, with the correct sign? If you can draw the two triangles and run the CAST check, you can.