Unit Circle Evaluator

Function

Exact Value

Decimal

sin θ

—

cos θ

—

tan θ

—

csc θ

—

sec θ

—

cot θ

—

ASTC — Signs by Quadrant

"All Students Take Calculus"

The mnemonic ASTC tells you which trig functions are positive in each quadrant. All others are negative.

Quadrant	Angles (°)	sin	cos	tan	Positive
I	0° – 90°	+	+	+	All
II	90° – 180°	+	−	−	Sin
III	180° – 270°	−	−	+	Tan
IV	270° – 360°	−	+	−	Cos

Reciprocal functions follow the same sign: csc shares sin's sign, sec shares cos's, cot shares tan's.

On the axes (0°, 90°, 180°, 270°, 360°), values are 0 or ±1 — no need to find a reference angle.

Exact Values from Special Triangles

30-60-90 and 45-45-90 triangles

All standard-angle exact values come from two triangles:

45-45-90: legs = 1, hyp = √2 → sin 45° = cos 45° = √2/2, tan 45° = 1
30-60-90: short leg = 1, long leg = √3, hyp = 2 → sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3; sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

Apply quadrant signs using ASTC to extend to all standard angles:

θ°	θ rad	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	√3/3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	undef
120°	2π/3	√3/2	−1/2	−√3
135°	3π/4	√2/2	−√2/2	−1
150°	5π/6	1/2	−√3/2	−√3/3
180°	π	0	−1	0

sin and cos of complementary angles are equal: sin(θ) = cos(90°−θ). That's why 30° and 60° swap sin and cos values.