Exact values for 30°, 45° and 60°

Fold a square along its diagonal and you get the exact value of sin 45°: one side over the diagonal, 1/√2, about 0.7071.

The 45-45-90 triangle

Most sines and cosines are long decimals, but three angles have values you can write exactly, and they turn up constantly in geometry, physics and design. The first comes from a square of side 1. Its diagonal cuts it into two right triangles with angles 45°, 45° and 90°. Both legs are 1, and by the Pythagorean theorem the hypotenuse is √(1² + 1²) = √2. OpenStax states the general pattern: the sides of a 45-45-90 triangle are in the relation s, s, √2 s.

Reading the ratios off this triangle: sin 45° = 1/√2 and cos 45° = 1/√2. Multiplying top and bottom by √2 gives the tidier form √2/2, about 0.7071. The two legs are equal, so tan 45° = 1/1 = 1. A roof or ramp that rises 1 for every 1 across has a 45° slope.

The 30-60-90 triangle

The second triangle comes from an equilateral triangle of side 2. All its angles are 60°. Drop a line from the top straight down to the middle of the base: it splits the triangle into two right triangles with angles 30°, 60° and 90°. Each has hypotenuse 2 and short side 1, half the base. The tall side is √(2² − 1²) = √3. OpenStax gives the pattern s, √3 s, 2s, with the short side opposite 30° and the long leg opposite 60°.

From 30°, the opposite side is 1 and the adjacent side is √3, so sin 30° = 1/2, cos 30° = √3/2 ≈ 0.8660 and tan 30° = 1/√3 = √3/3 ≈ 0.5774. From 60° the two legs swap roles: sin 60° = √3/2, cos 60° = 1/2 and tan 60° = √3 ≈ 1.7321.

Cofunctions

Notice that sin 30° equals cos 60°, and sin 60° equals cos 30°. That is no accident. The two acute angles of a right triangle add to 90°, and the side opposite one of them is adjacent to the other. So the sine of an angle is always the cosine of its complement: sin θ = cos(90° − θ). OpenStax calls this the cofunction identity. It means cos 34° and sin 56° are the same number without any calculation.

Exact values are worth knowing for two reasons. They let you finish a problem without a calculator, and they keep answers exact: a 30-60-90 triangle with short side 5 has a long leg of 5√3, which is exactly what it is, while 8.66 is already rounded. The same two triangles are the shapes of the two set squares in a geometry set, one with 45° corners and one with 30° and 60° corners, which is why drawings so often use these angles.