Question 1

What is sin(180°)?

Accepted Answer

sin(180°) = 0.

At 180°, the point on the unit circle is (−1, 0) — on the negative x-axis, directly opposite the starting point (1, 0). The y-coordinate is 0, so sin(180°) = 0.

Sine is zero at every integer multiple of 180°: sin(0°) = sin(180°) = sin(360°) = sin(−180°) = 0. These are the zero crossings of the sine function.

In a right triangle, 180° cannot be an interior angle — the triangle would degenerate. The unit circle definition extends sine to all angles including 180°.

Question 2

What is cos(180°)?

Accepted Answer

cos(180°) = −1.

At 180°, the unit circle point is (−1, 0). The x-coordinate gives cos(180°) = −1.

cos(180°) = −1 is the minimum value cosine can reach. Cosine oscillates between −1 and 1: it starts at 1 (at 0°), reaches −1 (at 180°), and returns to 1 (at 360°).

The negative value confirms we are in the second quadrant boundary — cos is negative in Q2 and Q3. 180° is the boundary point where cos reaches its most negative value.

Key identity: cos(180°) = −cos(0°) = −1. This follows from cos(180°−θ) = −cos(θ) with θ = 0.

Question 3

What is tan(180°)?

Accepted Answer

tan(180°) = 0.

tan = sin/cos = 0/(−1) = 0.

This is the same as tan(0°) = 0. Tangent has a period of 180°, meaning tan(θ + 180°) = tan(θ) for all angles. So tan(180°) = tan(0°) = 0.

Geometrically, at 180° the radius points to (−1, 0), which is on the x-axis. A radius pointing along the x-axis has slope = 0, confirming tan = 0.

Pattern: tan = 0 at 0°, 180°, 360° — whenever the point on the unit circle is on the x-axis (sin = 0). Tan is undefined at 90° and 270° — whenever the point is on the y-axis (cos = 0).

Question 4

Why are csc(180°) and cot(180°) undefined?

Accepted Answer

csc(180°) = 1/sin(180°) = 1/0 — undefined (division by zero).
cot(180°) = cos(180°)/sin(180°) = (−1)/0 — undefined (division by zero).

Both require dividing by sin(180°) = 0, which is undefined.

This happens at every angle where sin = 0: 0°, 180°, 360°, −180°... These are the vertical asymptotes of csc and cot.

Contrast with tan(180°) = 0 and sec(180°) = −1 — these are well-defined because they use cos(180°) = −1 in the denominator, which is not zero.

Question 5

What is the supplement identity and how does 180° relate to other angles?

Accepted Answer

The supplement identity says sin(180°−θ) = sin(θ) and cos(180°−θ) = −cos(θ).

This means angles that add up to 180° (supplementary angles) have the same sine but opposite cosines.

Examples:
- sin(150°) = sin(180°−30°) = sin(30°) = 1/2
- cos(150°) = −cos(30°) = −√3/2
- sin(120°) = sin(60°) = √3/2
- cos(120°) = −cos(60°) = −1/2
- sin(135°) = sin(45°) = √2/2
- cos(135°) = −cos(45°) = −√2/2

This is why Q2 angles (90° to 180°) have positive sine and negative cosine — sine mirrors Q1, cosine is negated.

180° also marks the start of the negative cosine region and the halfway point of a full rotation. In radians, 180° = π, which is why π is so fundamental in mathematics.

NAME	FORMULA	RESULT
Pythagorean identity at 180°	sin²(180°)+cos²(180°) = 0²+(−1)² = 0+1	= 1 ✓
Tan(180°)	tan(180°) = sin(180°)/cos(180°) = 0/(−1)	= 0
Csc(180°) undefined	csc(180°) = 1/sin(180°) = 1/0	= Undefined
Sec(180°)	sec(180°) = 1/cos(180°) = 1/(−1)	= −1
Cot(180°) undefined	cot(180°) = cos(180°)/sin(180°) = (−1)/0	= Undefined
Supplement identity	sin(180°−θ) = sin(θ)	e.g. sin(150°) = sin(30°) = 1/2
Supplement identity (cos)	cos(180°−θ) = −cos(θ)	e.g. cos(150°) = −cos(30°) = −√3/2
Half rotation	180° = π radians = half a full turn	The point is diametrically opposite 0°

Sin Cos Tan of 180 Degrees — Exact Values

TRIG VALUES AT 180°

UNIT CIRCLE

FORMULAS & IDENTITIES

WORKED EXAMPLE

FREQUENTLY ASKED QUESTIONS

RELATED CALCULATORS

MORE TRIGONOMETRY CALCULATORS

Function	Exact Value	Decimal
sin(180°)	0	0
cos(180°)	−1	-1
tan(180°)	0	0
csc(180°)	Undefined	—
sec(180°)	−1	-1
cot(180°)	Undefined	—

°	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