Sin 180 Degrees : Exact Value 0 and How to Find It

What Does Sin 180 Degrees Mean?

Sine is one of the three trigonometric ratios. For an acute angle it is opposite over hypotenuse, but $180^\circ$ collapses any triangle flat, so the ratio picture breaks down. The unit circle is the home definition here.

On the unit circle (radius $1$, centred at the origin), the sine of an angle is the y-coordinate of the point the radius reaches. Rotating a full $180^\circ$ from the positive x-axis lands exactly on the negative x-axis, at the point $(-1, 0)$. The y-coordinate is $0$, so $\sin 180^\circ = 0$. The x-coordinate is $-1$, which is why $\cos 180^\circ = -1$; the same point gives both sin, cos, and tan at $180^\circ$.

Where Does Sin 180 Degrees Show Up?

An alternating-current voltage modelled as $\sin\theta$ crosses zero at $180^\circ$ of phase - the exact midpoint of its cycle - which is why $\sin 180^\circ = 0$ describes a real zero-crossing in every power line. A pendulum swung through a half turn, or a projectile whose launch and landing lie on a level line, both hit the "$180^\circ$ later" mark where the vertical sine contribution vanishes. In wave physics, two signals $180^\circ$ out of phase cancel, and the sine of $180^\circ$ being zero is the arithmetic behind that cancellation, a staple of the applications of trigonometry.

Standard-Angle Reference Table

An angle of $180^\circ$ is a straight angle - the terminal side points in the exact opposite direction to where it started, along the negative x-axis. It is a quadrantal angle: it sits on an axis rather than inside a quadrant. Here are the standard angles from $0^\circ$ through $180^\circ$ in degrees and radians.

The sine starts at $0$, rises to $1$ at $90^\circ$, and returns to $0$ at $180^\circ$. So $\sin 0^\circ$ and $\sin 180^\circ$ are the same value, reached from opposite directions along the horizontal axis. The full set lives in the trigonometric table. Because $180^\circ = \pi$ radians, this article's radian twin, the value of sin pi, leads from the unit-circle and radian framing instead of the degree computation here.

How Do You Find The Exact Value Of Sin 180 Degrees?

Three routes all land on $0$: the unit circle reads it off directly, an identity derives it, and the sine curve confirms it.

Method 1: The unit circle.

The point at $180^\circ$ is $(-1, 0)$, and sine is the y-coordinate.

$$\sin 180^\circ = 0$$

Method 2: The supplementary-angle identity.

The identity $\sin(180^\circ - \theta) = \sin\theta$ holds for every angle. Set $\theta = 0^\circ$:

$$\sin 180^\circ = \sin(180^\circ - 0^\circ) = \sin 0^\circ = 0$$

Since $\sin 0^\circ = 0$, the supplement $\sin 180^\circ$ is also $0$. This is the mirror fact from the table: the sine curve is symmetric about $90^\circ$, so its two ends, $0^\circ$ and $180^\circ$, share the value $0$.

Method 3: The decimal check.

A calculator in degree mode returns $\sin(180) = 0$ exactly. In radian mode you must enter $\pi \approx 3.14159$, not $180$, or the calculator computes $\sin(180 \text{ rad}) \approx -0.801$, a different angle entirely.

Examples Of Sin 180 Degrees

Example 1

Evaluate $7\sin 180^\circ + 3$.

$$7\sin 180^\circ + 3 = 7 \times 0 + 3 = 3$$

Example 2

Simplify $\dfrac{\sin 180^\circ}{\cos 180^\circ}$, which is $\tan 180^\circ$.

Wrong attempt. A student sees a fraction with $\sin 180^\circ = 0$ in it and writes "undefined," assuming any trig fraction at $180^\circ$ blows up.

That confuses this with $\tan 90^\circ$. Here the denominator is $\cos 180^\circ = -1$, which is not zero, so nothing is undefined.

Correct. Substitute both values:

$$\tan 180^\circ = \frac{\sin 180^\circ}{\cos 180^\circ} = \frac{0}{-1} = 0$$

A zero on top over a nonzero bottom is simply $0$. The tangent is only undefined when the cosine in the denominator is zero, and the reflex to call every $180^\circ$ fraction "undefined" is the slip to watch.

Example 3

A voltage is modelled by $V = 12\sin\theta$. Find $V$ at $\theta = 180^\circ$.

$$V = 12 \times \sin 180^\circ = 12 \times 0 = 0 \text{ volts}$$

The waveform is at a zero-crossing.

Example 4

Verify that $\sin 180^\circ = \sin 0^\circ$.

Both equal $0$. On the unit circle, $0^\circ$ sits at $(1, 0)$ and $180^\circ$ at $(-1, 0)$; the x-coordinates differ, but both y-coordinates are $0$, so the sines match.

Example 5

Express $\sin 180^\circ$ in radians and evaluate $\sin \pi$.

Since $180^\circ = \pi$ radians, $\sin \pi = \sin 180^\circ = 0$. The degree form and the radian form name one straight angle and one value.

Where Students Trip Up On Sin 180 Degrees

Mistake 1: Confusing sin 180° with cos 180°

Where it slips in: Reading the wrong coordinate off the point $(-1, 0)$.

Don't do this: Writing $\sin 180^\circ = -1$.

The correct way: Sine is the y-coordinate, which is $0$; the $-1$ is the x-coordinate, so it is $\cos 180^\circ$. When a $180^\circ$ answer comes out as $-1$, cosine got read where sine was wanted.

Mistake 2: Getting −0.801 from a calculator

Where it slips in: A calculator left in radian mode, with $180$ typed in.

Don't do this: Trusting $\sin(180) \approx -0.801$ from radian mode.

The correct way: For degrees, switch to degree mode, or enter $\pi$ for the radian value. The habit that prevents this is checking the mode indicator before every trig entry.

Mistake 3: Thinking sin 180° must be undefined

Where it slips in: Assuming a quadrantal angle produces an undefined ratio, the way $\tan 90^\circ$ does.

Don't do this: Reporting $\sin 180^\circ$ as "undefined" or "no value."

The correct way: Sine is defined for every angle; only tangent, secant, cosecant, and cotangent go undefined where their denominator is zero. $\sin 180^\circ$ is a clean $0$.

Key Takeaways

• Sin 180 degrees equals $0$ exactly — the sine curve returns to zero at a straight angle.

• On the unit circle, $180^\circ$ sits at $(-1, 0)$, so the y-coordinate, and the sine, is $0$; the x-coordinate gives $\cos 180^\circ = -1$.

• In radians, $\sin 180^\circ = \sin \pi = 0$, and by symmetry $\sin 180^\circ = \sin 0^\circ$.

• The common slips are reading cosine's $-1$, radian-mode calculator errors, and calling a defined value "undefined."

• To go further with a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or math classes online.

Practice These Before Moving On

1. Evaluate $\sin 180^\circ + \cos 180^\circ + \sin 90^\circ$.

2. Simplify $5\sin 180^\circ - 2\cos 180^\circ$.

3. A signal is $A\sin\theta$; explain in one line why it reads zero at both $\theta = 0^\circ$ and $\theta = 180^\circ$.

Want a live Bhanzu trainer to walk through more sin 180 degrees problems? Book a free demo class.

Read More

• Cofunction identities — how sine and cosine swap at complementary angles.

• Sin 120 degrees — a second-quadrant standard angle equal to $\frac{\sqrt{3}}{2}$.

• Sin 150 degrees — a second-quadrant standard angle equal to $\frac{1}{2}$.

• Pythagorean identities — how $\sin 180^\circ = 0$ forces $\cos 180^\circ = \pm 1$.

• Sum and difference formulas — the identities behind $\sin(180^\circ - \theta) = \sin\theta$.