Question 1

What is sin(360°)?

Accepted Answer

sin(360°) = 0.

At 360°, the point on the unit circle has completed a full rotation and returned to (1, 0) — exactly where it started at 0°. The y-coordinate is 0, so sin(360°) = 0.

This is a direct consequence of the periodicity of sine: sin(θ + 360°) = sin(θ) for all θ.

So sin(360°) = sin(0°) = 0. The values are identical because the geometric position on the unit circle is identical.

Question 2

What is cos(360°)?

Accepted Answer

cos(360°) = 1.

At 360°, the unit circle point is (1, 0). The x-coordinate gives cos(360°) = 1.

Cosine has period 360°: cos(θ + 360°) = cos(θ). So cos(360°) = cos(0°) = 1.

cos(360°) = 1 is the maximum value of cosine, shared with cos(0°), cos(720°), cos(−360°), and every full rotation. The point (1, 0) on the positive x-axis is the 'home base' of the unit circle.

Question 3

Why are sin(360°) and cos(360°) the same as sin(0°) and cos(0°)?

Accepted Answer

Because 360° is exactly one full rotation. After rotating 360° counterclockwise from (1, 0), you return to the exact same point — (1, 0).

The unit circle definition of trig functions depends only on the position of the point, not the total angle traversed. Since 360° and 0° land on the same point, all trig values are identical.

Formal statement: all six trig functions are periodic with period 360° (2π):
- sin(θ + 360°) = sin(θ)
- cos(θ + 360°) = cos(θ)
- tan(θ + 180°) = tan(θ) (tan has period 180°)

This means:
- sin(360°) = sin(0°) = 0
- cos(360°) = cos(0°) = 1
- tan(360°) = tan(0°) = 0
- sec(360°) = sec(0°) = 1
- csc(360°) = csc(0°) = undefined
- cot(360°) = cot(0°) = undefined

Question 4

What is 360 degrees in radians?

Accepted Answer

360° = 2π radians ≈ 6.28318530 radians.

Conversion: radians = degrees × π/180 = 360 × π/180 = 2π.

Why 2π? The circumference of the unit circle is 2π × radius = 2π × 1 = 2π. One full rotation travels the full circumference, which is 2π radians.

Key radian conversions:
- 0° = 0
- 90° = π/2
- 180° = π
- 270° = 3π/2
- 360° = 2π

2π ≈ 6.2832 is the period of sine and cosine. Euler's famous identity e^(i·2π) = 1 expresses this: raising e to the imaginary power 2π brings you exactly back to 1 — a full rotation in the complex plane.

Question 5

How many degrees are in a full circle and why?

Accepted Answer

A full circle = 360 degrees. The choice of 360 is historical, not mathematical.

Origin: Ancient Babylonians used a base-60 number system and observed approximately 360 days in a year. They divided the circle into 360 equal parts, each corresponding to roughly one day of the sun's apparent motion.

Why 360 is practical:
- 360 has many divisors: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
- This makes it easy to divide a circle into equal parts without fractions
- 360/6 = 60° (hexagon), 360/4 = 90° (square), 360/3 = 120° (triangle)

Alternatives:
- Radians (2π per circle): mathematically natural, used in calculus
- Gradians (400 per circle): right angle = 100 grad, used in surveying
- Turns (1 per circle): occasionally used in programming

For mathematics, radians are preferred. For everyday angles, degrees remain standard.

°	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	Undef
120°	√3/2	−1/2	−√3
135°	√2/2	−√2/2	−1
150°	1/2	−√3/2	−1/√3
180°	0	−1	0
270°	−1	0	Undef
360°	0	1	0

NAME	FORMULA	RESULT
Pythagorean identity at 360°	sin²(360°)+cos²(360°) = 0²+1² = 0+1	= 1 ✓
Periodicity of sine	sin(360°) = sin(360°−360°) = sin(0°) = 0	Period = 360°
Periodicity of cosine	cos(360°) = cos(0°) = 1	Period = 360°
Tan(360°)	tan(360°) = sin(360°)/cos(360°) = 0/1	= 0 (same as tan(0°) and tan(180°)
Csc(360°) undefined	csc(360°) = 1/sin(360°) = 1/0	= Undefined
Sec(360°)	sec(360°) = 1/cos(360°) = 1/1	= 1
Cot(360°) undefined	cot(360°) = cos(360°)/sin(360°) = 1/0	= Undefined
Full rotation	360° = 2π radians = 1 full turn	Returns to exact starting position

Sin Cos Tan of 360 Degrees — Exact Values

TRIG VALUES AT 360°

UNIT CIRCLE

FORMULAS & IDENTITIES

WORKED EXAMPLE

FREQUENTLY ASKED QUESTIONS

RELATED CALCULATORS

MORE TRIGONOMETRY CALCULATORS

Function	Exact Value	Decimal
sin(360°)	0	0
cos(360°)	1	1
tan(360°)	0	0
csc(360°)	Undefined	—
sec(360°)	1	1
cot(360°)	Undefined	—