Cos 180 Degrees = −1 — Value, Unit Circle, Radians

Quick Answer:

• Result: $\cos 180° = -1$

• Notation: $\cos\pi = -1$ (radians)

• Method shown: unit circle — the $x$-coordinate at the half-turn

• Approximate value: $-1$ (exact, no rounding)

• Exact form: $-1$

A half-turn around the unit circle — exactly $180°$, or $\pi$ radians — carries you from the starting point $(1, 0)$ to the point directly opposite, $(-1, 0)$. Cosine reads the $x$-coordinate of where you land, and that $x$-coordinate is $-1$. This is the most negative value cosine ever reaches, the bottom of the cosine wave.

Quick rReference Table — Cosine of Standard Angles

This table lists cosine at the standard angles around the circle, in both degrees and radians, with $180°$ highlighted.

Cosine is positive in quadrants I and IV, negative in II and III. At $180°$ the angle sits exactly on the negative $x$-axis, where the $x$-coordinate bottoms out at $-1$.

Where cos 180 Degrees Appears

The value $\cos 180° = -1$ turns up wherever something fully reverses direction. In alternating-current circuits, a voltage and current that are $180°$ out of phase are exact negatives of each other at every instant, because $\cos 180° = -1$ scales one to the opposite of the other. Active noise-cancelling headphones use the same idea, generating a sound wave $180°$ out of phase with incoming noise so the two cancel. It is also the heart of Euler's identity, $e^{i\pi} = -1$, which is just $\cos\pi + i\sin\pi$ with the sine term vanishing.

What is cos 180 Degrees?

Cosine of an angle is, on the unit circle, the $x$-coordinate of the point reached by rotating that angle counterclockwise from the positive $x$-axis. For $180°$, that rotation is exactly half a full turn.

A half-turn lands on $(-1, 0)$ — the leftmost point of the circle. The $x$-coordinate there is $-1$, so $\cos 180° = -1$. The same angle written in radians is $\pi$, which is why $\cos 180°$ and $\cos\pi$ are the same number.

How To Find The Value of cos 180 Degrees

Method 1 — Unit circle

Rotate $180°$ counterclockwise from $(1, 0)$. You land on the point directly opposite, $(-1, 0)$.

Cosine is the $x$-coordinate of that point.

Final answer: $\cos 180° = -1$.

Method 2 — Cosine addition formula

Split $180°$ as $90° + 90°$ and apply $\cos(A + B) = \cos A\cos B - \sin A\sin B$:

$$\cos(90° + 90°) = \cos 90°\cos 90° - \sin 90°\sin 90°$$

$$= (0)(0) - (1)(1) = -1$$

Final answer: $\cos 180° = -1$.

Method 3 — Supplementary-angle identity

Use $\cos(180° - \theta) = -\cos\theta$ with $\theta = 0°$:

$$\cos 180° = \cos(180° - 0°) = -\cos 0° = -(1) = -1$$

Common mistakes with cos 180 degrees

Mistake 1: Confusing cos 180° with sin 180°

Where it slips in: At $180°$ the unit-circle point is $(-1, 0)$, and the two coordinates get swapped.

Don't do this: Writing $\cos 180° = 0$ (that is $\sin 180°$, the $y$-coordinate).

The correct way: Cosine is the $x$-coordinate, so $\cos 180° = -1$; sine is the $y$-coordinate, so $\sin 180° = 0$.

Mistake 2: Calculator in degree vs radian mode

Where it slips in: Typing cos(180) while the calculator is in radian mode.

Don't do this: Reading $\cos(180 \text{ rad}) \approx -0.598$ and reporting it as $\cos 180°$.

The correct way: Set the calculator to degree mode for $\cos 180°$, or enter $\cos\pi$ in radian mode. Both give $-1$.

Mistake 3: Treating cos 180° as positive

Where it slips in: Recalling the magnitude 1 but dropping the sign.

Don't do this: Writing $\cos 180° = 1$.

The correct way: $180°$ is in the second-quadrant boundary on the negative $x$-axis, where cosine is negative, so $\cos 180° = -1$.

Conclusion

• Cos 180 degrees equals $-1$ — the cosine of a half-turn, written $\cos\pi$ in radians.

• The half-turn lands on $(-1, 0)$, the leftmost point of the unit circle; cosine reads that $x$-coordinate.

• Three routes agree: unit circle, the addition formula on $90° + 90°$, and the supplementary-angle identity.

• The most common slip is swapping cosine and sine — $\cos 180° = -1$, $\sin 180° = 0$.

Quick self-check — try these

1. Evaluate $\cos 180° + \sin 180°$.

2. Write $\cos 180°$ in radians and state its value.

3. Explain in one sentence why $\cos 180°$ is negative.

If #1 didn't give $-1$, recheck the coordinates of the point at $180°$. Want a live Bhanzu trainer to walk through more unit-circle problems? Book a free demo class — online globally.