// maths › Trigonometric Identities
Read sin, cos and tan as exact surds at every special angle — 0, 30°, 45°, 60°, 90° and their reflections in all four quadrants — derived from the special triangles, the unit circle and the CAST rule, in degrees or radians.
sin 30°=1/2, cos 30°=√3/2, sin 45°=cos 45°=√2/2, sin 60°=√3/2
A mind behind this: Pythagoras of Samos c. 570 – c. 495 BC
The standard special angles: 0, 30°, 45°, 60° and 90°, plus all of their reflections around the circle (120°, 135°, 150°, 180° … up to 360°). Their sin, cos and tan can be written exactly as surds — like √3/2 or √2/2 — with no calculator needed.
From two set-square triangles and the unit circle. The 45-45-90 triangle (sides 1 : 1 : √2) gives the 45° values, and the 30-60-90 triangle (sides 1 : √3 : 2) gives the 30° and 60° values. The axis angles (0, 90°, 180°, 270°) are read straight off the unit circle.
Take θ = 150°. It sits in quadrant 2 with a reference angle of 30°, so the magnitudes match 30°: sin = 1/2, cos = √3/2, tan = √3/3. CAST says only sin is positive in quadrant 2, so sin 150° = 1/2, cos 150° = −√3/2 and tan 150° = −√3/3. The same angle in radians is 150° = 5π/6 ≈ 2.6180 rad, and the toggle re-derives the working in radians.
CAST tells you which functions are positive in each quadrant, reading anticlockwise from quadrant 4: All in Q1, Sin in Q2, Tan in Q3, Cos in Q4. You find the value from the reference angle's triangle, then CAST fixes the sign.
tan θ = sin θ / cos θ, and at 90° and 270° the cosine is 0, so the division is undefined. The calculator flags this rather than returning a misleading number.