Sin Cos Tan Table: Values for 0°, 30°, 45°, 60°, 90°

The sin cos tan table gives the exact values of sin, cos and tan for the five angles that appear in almost every textbook problem — 0°, 30°, 45°, 60° and 90°. Sin 30° = 0.5, cos 30° = 0.8660, tan 30° = 0.5774; sin 45° = cos 45° = 0.7071, tan 45° = 1; sin 60° = 0.8660, cos 60° = 0.5, tan 60° = 1.7321. At 0° and 90° the values are 0 and 1, and tan 90° is undefined.

This table is the single most useful thing to memorise in trigonometry. Exam questions are built around these five angles precisely because their values are exact — no calculator needed. This post gives you the full table, shows where every value comes from, and teaches the patterns that make it stick.

Check any value instantly with a free trigonometry calculator, which returns sin, cos and tan for any angle.

The Sin Cos Tan Table

Read the table down each column and two patterns jump out. First, the sin column climbs 0, 0.5, 0.7071, 0.8660, 1 while the cos column falls in exactly the reverse order — 1, 0.8660, 0.7071, 0.5, 0. That is not a coincidence: it is the co-function rule, sin θ = cos(90° − θ). Sin 30° is cos 60°, and both are 0.5. Second, tan 45° = 1 sits in the middle like a hinge — tan is below 1 before 45° and above 1 after it, because tan is opposite over adjacent, and at 45° the two legs are equal.

Where the Values Come From: Two Triangles

The whole table comes from just two triangles. Once you can draw them, you never need to memorise the table blindly again.

The 30-60-90 triangle (half an equilateral triangle). Start with an equilateral triangle of side 2 — all angles 60°. Cut it in half down the middle and you get a right triangle with angles 30°, 60° and 90°, hypotenuse 2, short leg 1 (half of 2) and long leg √3 (from Pythagoras: √(2² − 1²) = √3 = 1.7321). Now read the ratios off it:

• Sin 30° = opposite/hypotenuse = 1/2 = 0.5
• Cos 30° = adjacent/hypotenuse = √3/2 = 0.8660
• Sin 60° = √3/2 = 0.8660 (the same numbers, swapped — the long leg is now opposite)
• Cos 60° = 1/2 = 0.5

That is the entire 30° and 60° rows of the table, derived from one cut. And it explains the 30-60-90 shortcut: with hypotenuse 10, the short leg is 5 and the long leg is 5√3 = 8.6603.

The 45-45-90 triangle (half a square). Start with a square of side 1 and cut it along a diagonal. You get a right triangle with angles 45°, 45° and 90°, both legs 1, and hypotenuse √2 (from Pythagoras: √(1² + 1²) = √2 = 1.4142). Read the ratios:

• Sin 45° = 1/√2 = √2/2 = 0.7071
• Cos 45° = 1/√2 = √2/2 = 0.7071
• Tan 45° = 1/1 = 1

Both legs are equal, so sin and cos are the same number, and tan is exactly 1. That is the whole 45° row.

The 0° and 90° rows are the limits. At 0° the triangle flattens completely: the opposite side shrinks to 0, so sin 0° = 0, while the adjacent side becomes the whole hypotenuse, so cos 0° = 1 — and tan 0° = 0/1 = 0. At 90° it is the reverse: sin 90° = 1, cos 90° = 0, and tan 90° = 1/0, which is division by zero — undefined, not 0 and not 1.

The √n/2 Pattern: Memorise Sin in 10 Seconds

Here is the trick that makes the sin column unforgettable. Write the sin values as square roots over 2:

• sin 0° = √0/2 = 0
• sin 30° = √1/2 = 0.5
• sin 45° = √2/2 = 0.7071
• sin 60° = √3/2 = 0.8660
• sin 90° = √4/2 = 1

The numbers under the square root run 0, 1, 2, 3, 4 — just count up. The cos column is the same pattern in reverse: √4/2, √3/2, √2/2, √1/2, √0/2. Learn one column and you have both. Then tan is sin divided by cos, so tan 30° = 0.5/0.8660 = 0.5774, tan 45° = 0.7071/0.7071 = 1, and tan 60° = 0.8660/0.5 = 1.7321 — each one falls out of the two columns you already know.

Using the Table: Four Worked Examples

Example 1 — the opposite side from the angle. A right triangle has hypotenuse 10 and angle 30°. The opposite side is 10 × sin 30° = 10 × 0.5 = 5. The adjacent side is 10 × cos 30° = 10 × 0.8660 = 8.6603. No calculator needed — both values are straight from the table.

Example 2 — the angle from the sides. A right triangle has opposite 5 and hypotenuse 10. Sin θ = 5/10 = 0.5, and the table says sin θ = 0.5 at exactly 30° — so θ = 30°, no inverse trig needed. This is how exam questions are designed: the ratio lands exactly on a table value.

Example 3 — the height from the angle. A ramp rises at 60° and runs 5 units horizontally. Its height is 5 × tan 60° = 5 × 1.7321 = 8.6605 units (8.6603 to 4 decimal places — the tiny difference is rounding in 1.7321). The hypotenuse is then 10, since this is the 30-60-90 triangle with short leg 5.

Example 4 — the 3-4-5 triangle is not in the table. Opposite 3, hypotenuse 5 gives sin θ = 0.6 — and 0.6 is not in the table, so θ is not a special angle. It is arcsin(0.6) = 36.87°, with cos θ = 0.8 and tan θ = 3/4 = 0.75 (not 4/5 — the adjacent side is 4, from 5² − 3² = 16). The table covers the five special angles; for anything else, use inverse trig — a free trigonometry calculator gives the angle from any ratio.

Beyond the Five Angles: 15° and 75°

The table stops at five angles, but the same ideas extend further. The values for 15° and 75° also have exact forms, and they show the co-function symmetry again:

Sin 15° = cos 75° = 0.2588, and sin 75° = cos 15° = 0.9659 — the same mirror pattern as 30° and 60°. These values are worth knowing if your course covers them, but they are not in the standard five-angle table, so check whether your exam expects them before memorising.

Checking Your Table Values

Every row of the table must pass the identity sin²θ + cos²θ = 1. Run it on the 30° row: 0.5² + 0.8660² = 0.25 + 0.75 = 1 ✓. On the 45° row: 0.7071² + 0.7071² = 0.5 + 0.5 = 1 ✓. On the 60° row: 0.8660² + 0.5² = 0.75 + 0.25 = 1 ✓. If a value you memorised fails this check, it is wrong — the identity is the table’s built-in proofreader.

The second check is tan = sin/cos. Tan 30° = 0.5/0.8660 = 0.5774 ✓, tan 60° = 0.8660/0.5 = 1.7321 ✓. If your tan does not equal sin divided by cos, one of the three is wrong.

Common Mistakes With the Sin Cos Tan Table

Mixing up the 30° and 60° rows. Sin 30° = 0.5 but sin 60° = 0.8660 — they swap. The memory aid is the triangle: the side opposite 30° is the short leg (half the hypotenuse), so sin 30° is the small number, 0.5. The side opposite 60° is the long leg, so sin 60° is the big number, 0.8660.

Writing tan 90° as 0 or 1. Tan 90° is undefined — cos 90° = 0, and division by zero has no answer. If a calculator shows an enormous number (like 1.6 × 10¹⁶) for tan 90°, that is its way of saying “undefined”.

Using radians by accident. The table is in degrees. Sin(30 radians) = −0.9880, which is nothing like 0.5. If a value from your calculator disagrees with the table, check the angle mode first.

Rounding √3/2 to 0.87. Cos 30° = 0.8660 — rounded to 0.87 it still looks right, but in a chain of calculations the error grows. Keep 4 decimal places (0.8660, 0.5774, 1.7321) until the final answer.

Forgetting the table works both ways. The table answers “what is sin 30°?” (0.5) and “which angle has sin 0.5?” (30°). Exam questions use both directions, and the second one is just reading the table backwards.

Frequently Asked Questions

What is sin 30° cos 30° tan 30°?
Sin 30° = 0.5, cos 30° = 0.8660, tan 30° = 0.5774. The exact values are 1/2, √3/2 and 1/√3, from the 30-60-90 triangle (half an equilateral triangle of side 2).

What is the value of tan 45°?
1. At 45° the opposite and adjacent sides are equal (the 45-45-90 triangle is half a square), so tan 45° = opposite/adjacent = 1. Sin 45° and cos 45° are both √2/2 = 0.7071.

What is sin 60° and cos 60°?
Sin 60° = 0.8660 (√3/2) and cos 60° = 0.5 (1/2). They are the 30° values swapped — the co-function rule, sin θ = cos(90° − θ). Tan 60° = √3 = 1.7321.

How do you memorise the sin cos tan table?
Use the √n/2 pattern for sin: √0/2, √1/2, √2/2, √3/2, √4/2 for 0°, 30°, 45°, 60°, 90°. Cos is the same pattern backwards, and tan is sin divided by cos. Learn where the values come from — the two triangles — and the table rebuilds itself.

Why is tan 90° undefined?
Because cos 90° = 0, and tan = sin/cos means dividing by zero. Tan 90° = 1/0 has no value. The same applies to any angle where cos θ = 0.

What is sin cos tan of 0 degrees?
Sin 0° = 0, cos 0° = 1, tan 0° = 0. The triangle flattens: the opposite side is 0 and the adjacent side is the whole hypotenuse.

Try It Yourself

Print the table, cover one column, and fill it in from the √n/2 pattern — then check yourself with the free sin cos tan table tool, which returns sin, cos and tan for any angle, including the five special angles. For the missing side behind any table value, the Pythagorean Theorem Calculator finds it from a² + b² = c², and the Triangle Calculator solves whole triangles — right or not — with the law of sines and cosines.