Special Triangles 30-60-90 and 45-45-90 and Trig Value Tables

The 30°-60°-90° and 45°-45°-90° triangles have side ratios 1 : √3 : 2 and 1 : 1 : √2, and they give the exact trigonometric values for 30°, 45° and 60° without a calculator. In brief: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 = √3/3, sin 45° = cos 45° = √2/2, tan 45° = 1, sin 60° = √3/2, cos 60° = 1/2 and tan 60° = √3.

These values appear so often that it is worth remembering them, and they can be rebuilt in seconds from two simple triangles. Together with the values for 0° and 90°, they form the standard table used in algebra, geometry and calculus. This guide shows where the ratios come from, how to use them for exact answers and gives a complete table of exact and decimal values.

Where does the 45°-45°-90° triangle come from?

Cut a square along its diagonal. Each half is a right isosceles triangle with two equal legs of length 1 and a hypotenuse of √(1 + 1) = √2 by the Pythagorean theorem. The acute angles are 45° each, since the diagonal bisects the right angles. The sides are in the ratio 1 : 1 : √2. For legs of length x, the hypotenuse is x√2, and for hypotenuse h, each leg is h/√2 = h√2/2.

Therefore sin 45° = 1/√2 = √2/2 ≈ 0.7071, cos 45° = √2/2 and tan 45° = 1.

Where does the 30°-60°-90° triangle come from?

Cut an equilateral triangle with side 2 along an altitude. Each half is a right triangle with hypotenuse 2, the shortest leg 1 (half of the base) and the other leg √(4 − 1) = √3. The angles are 30° (at the top, half of 60°), 60° and 90°. The sides are in the ratio 1 : √3 : 2, with the shortest side opposite 30° and the longest (hypotenuse) opposite 90°.

Then sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 = √3/3 ≈ 0.5774, sin 60° = √3/2 ≈ 0.8660, cos 60° = 1/2 and tan 60° = √3 ≈ 1.7321.

What are the exact values for 0°, 30°, 45°, 60° and 90°?

For 0° and 90°, the values come from limiting cases of a right triangle (a degenerate triangle) and are confirmed on the unit circle. sin 0° = 0, cos 0° = 1, tan 0° = 0; sin 90° = 1, cos 90° = 0, and tan 90° is undefined, since the division by cos 90° = 0 is not possible.

How do you use special triangles in problems?

Example 1: exact side

A right triangle has an angle of 30° and the hypotenuse 12. The side opposite 30° is 12 · sin 30° = 6, and the adjacent side is 12 · cos 30° = 6√3 ≈ 10.39. By the ratio 1 : √3 : 2 with the factor 6, we get the same.

Example 2: 45°

A ramp makes an angle of 45° with the ground and is 10 m long. The height is 10 · sin 45° = 5√2 ≈ 7.07 m, equal to the horizontal distance.

Example 3: height of an equilateral triangle

An equilateral triangle with side a has the height a · sin 60° = a√3/2, so its area is (1/2) · a · a√3/2 = (√3/4)a². For a = 8 the height is 4√3 ≈ 6.93 and the area is 16√3 ≈ 27.71.

Example 4: evaluating an expression exactly

Compute sin 30° cos 60° + tan 45°: (1/2)(1/2) + 1 = 5/4. And sin²60° + cos²60° = 3/4 + 1/4 = 1.

Example 5: a diagonal path

A kite string of length 40 m makes an angle of 60° with the ground. The height of the kite is 40 sin 60° = 20√3 ≈ 34.64 m, and the horizontal distance is 40 cos 60° = 20 m.

Exact and decimal value table

How can you remember the table?

For sine, the values for 0°, 30°, 45°, 60° and 90° are √0/2, √1/2, √2/2, √3/2 and √4/2, which simplify to 0, 1/2, √2/2, √3/2 and 1. Cosine is the same list in reverse order. Tangent is sine divided by cosine. Another memory aid is the “hand trick”: hold up the left hand with the thumb as 0° and the little finger as 90°, and count the fingers on the left of the chosen finger to get the number under the root.

Where do these values come from?

The special triangles are as old as the Pythagorean theorem. The ratio 1 : 1 : √2 shows that the diagonal of a square is incommensurable with its side, a discovery attributed to the early Pythagoreans. Ptolemy’s table of chords included exact values for the angles related to the regular hexagon, square and pentagon.

Where are they used today?

Carpenters use 45° miter cuts and 30°-60° set squares. Engineers split forces into components using these angles, and architects design roofs with a pitch of 30° or 45°. Exact values are also needed in exams and in calculus, where answers are given in terms of √2 and √3, and in computer graphics for rotations through multiples of 30° and 45°.

Which mistakes are common?

Mixing up 30° and 60°

The shortest side is opposite the smallest angle. The side opposite 30° is 1 (half of the hypotenuse) and the side opposite 60° is √3.

Forgetting to rationalize

tan 30° = 1/√3 is equal to √3/3. Both are correct, but many courses require the rationalized form.

Using decimals when the problem asks for exact values

Leave answers as √2/2 or √3/2 when exact values are required.

Applying the 1 : 1 : √2 ratio to the wrong sides

The factor √2 belongs to the hypotenuse, not to a leg.