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# Value of Sin pi — sin(π) in Radians & Unit Circle

TL;DR

The value of sin pi is 0. In radians, π corresponds to 180° — half a full rotation around the unit circle. The point on the unit circle at this angle is (−1,0). Since sine reads the y-coordinate of that point, sin⁡π=0.

BT

[Bhanzu Team](/content/authors/bhanzu-team/index.html) Last updated on May 20, 2026 9 min read

## A Half Rotation Lands You Back on the Axis

A full rotation around a circle returns you to your starting point. Half a rotation — π radians, or 180° — lands you on the _opposite side_, but if your starting point was on the horizontal axis, your _vertical position_ is the same: zero. That's the entire story of sin⁡π=0: the half-rotation keeps you on the horizontal axis, and sine reads the vertical coordinate.

## What Is sin pi?

sin⁡π is the sine of the angle π radians, equivalent to sin⁡(180°). Its value is:

sin⁡π=0

This is one of the foundational exact values in trigonometry — alongside sin⁡0=0, sin⁡(π/2)=1, sin⁡(π/6)=1/2, sin⁡(π/4)=√2/2, and sin⁡(π/3)=√3/2.

## Three Methods to Find sin pi

### Method 1 — Unit Circle

On the unit circle (radius 1, centred at the origin), the point at angle θ has coordinates (cos⁡θ,sin⁡θ).

The angle π corresponds to half a rotation — landing on the point (−1,0).

So sin⁡π=0 (the y-coordinate) and cos⁡π=−1 (the x-coordinate).

### Method 2 — From the Sine Graph

The sine curve has these zeros: x=0,π,2π,3π,… and x=−π,−2π,… . More precisely, sin⁡x=0 for every integer multiple of π.

Reading the graph at x=π: sin⁡π=0.

### Method 3 — Using the Supplementary-Angle Identity

The identity sin⁡(π−θ)=sin⁡θ holds for every θ. Setting θ=0:

sin⁡π=sin⁡(π−0)=sin⁡0=0.

Three independent methods, same answer. The agreement is the whole point of identities — they make the value un-mistakable.

## Unit Circle Sine Reference — All Standard Angles

The complete sine values around the unit circle, organised by quadrant:

| Angle (radians) | Angle (degrees) | sin⁡θ | Quadrant |
| --- | --- | --- | --- |
| 0 | 0° | 0 | Axis |
| π/6 | 30° | 1/2 | I |
| π/4 | 45° | √2/2 | I |
| π/3 | 60° | √3/2 | I |
| π/2 | 90° | 1 | Axis |
| 2π/3 | 120° | √3/2 | II |
| 3π/4 | 135° | √2/2 | II |
| 5π/6 | 150° | 1/2 | II |
| **π** | **180°** | **0** | **Axis** |
| 7π/6 | 210° | −1/2 | III |
| 5π/4 | 225° | −√2/2 | III |
| 4π/3 | 240° | −√3/2 | III |
| 3π/2 | 270° | −1 | Axis |
| 5π/3 | 300° | −√3/2 | IV |
| 7π/4 | 315° | −√2/2 | IV |
| 11π/6 | 330° | −1/2 | IV |
| 2π | 360° | 0 | Axis |

### **Pattern lock-in.** 
Sine is positive in quadrants I and II (upper half of the circle), negative in III and IV (lower half), and zero on the horizontal axis (0, π, 2π). The four "axis" angles — 0, π/2, π, 3π/2 — give the four "extreme" sine values: 0, 1, 0, −1.

## sin pi in Terms of Other Trigonometric Functions

The full set of trig function values at θ=π:

sin⁡π=0, cos⁡π=−1, tan⁡π=0, csc⁡π=undefined (division by zero), sec⁡π=−1, cot⁡π=undefined (division by zero).

Two of the six are undefined because sin⁡π=0 appears in the denominator. The angle π is a "node" — a point where the wave passes through zero.

## Three Worked Examples — Quick, Standard, Stretch

### Quick

**Evaluate sin⁡π+cos⁡π.**

sin⁡π=0 and cos⁡π=−1.

sin⁡π+cos⁡π=0+(−1)=−1.

### A Common Slip Worth Walking Through — Standard Example

**Evaluate sin⁡(2π/3)+sin⁡π+sin⁡(7π/6).**

_The wrong path._ A student reasons: "sin⁡(2π/3) is in quadrant II — positive. sin⁡π=0. sin⁡(7π/6) is in quadrant III — negative. So the answer is sin⁡(2π/3)+0+sin⁡(7π/6)=sin⁡(60°)−sin⁡(30°)."

Wait — what are the reference angles?

**Sanity check.** 2π/3=120°, so its reference angle is π−2π/3=π/3=60°. That's right. 7π/6=210°, so its reference angle is 7π/6−π=π/6=30°. That's right. So sin⁡(7π/6)=−sin⁡(π/6)=−1/2, and the answer is 3/2+0+(−1/2)=(3−1)/2.

### Stretch

**A pendulum's vertical displacement is y(t)=0.3sin⁡(πt). Where is the pendulum at t=1,2,3 seconds?**

At t=1: y(1)=0.3sin(π)=0.3⋅0=0 m.

At t=2: y(2)=0.3sin(2π)=0.3⋅0=0 m.

At t=3: y(3)=0.3sin(3π)=0.3⋅0=0 m.

At every integer value of t, the pendulum is at y=0 — its equilibrium position. The period is 2 seconds (so the pendulum returns to y=0 every 1 second).

## Where sin pi = 0 Quietly Powers the World

sin⁡π=0 may look trivial, but its periodic cousin sin⁡(nπ)=0 is the foundation of every oscillation, wave, and Fourier-series analysis.

- **Standing waves on a string.** 
- **AC voltage zero-crossings.** 
- **Fourier series.** 
- **Quantum mechanics.**

## Where Things Go Sideways — Common Mistakes

### **Mistake 1: Confusing sin⁡π with sin⁡(π°)** 
**Don't do this:** Computing sin⁡π in degree mode.

### **Mistake 2: Reading sin⁡π as sin⁡sin × π**  
**Don't do this:** Writing "sin⁡π=π" because the symbols look juxtaposed.

### **Mistake 3: Thinking sin⁡ of any small fraction of π must be small** 
**Don't do this:** Assuming sin⁡(3π/7)≈0 because "3π/7 is close to π".

## Key Takeaways

- sin⁡π=0 — the sine of the angle 180° measured in radians.
- The angle π lands on the point (−1,0) on the unit circle.
- Three independent methods give the same answer: unit circle, sine graph, supplementary identity.
- Always check the calculator mode.
- sin⁡(nπ)=0 for every integer n — the engine behind standing waves, AC zero-crossings, Fourier series, and quantum mechanics.

## Take Sin pi for a Test Drive — Three Problems

1. Evaluate sin⁡π+sin⁡(2π)+sin⁡(3π).
2. Find sin⁡(5π/4) using the unit-circle table above.
3. Show that sin⁡π⋅cos⁡(π/2)+cos⁡π⋅sin⁡(π/2)=-1.

If #1 didn't give you 0, re-check that sine is zero at every integer multiple of π. If #2 didn't give you −√2/2, look at the table — 5π/4 is in quadrant III.

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