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.
Angle (degrees) | Angle (radians) | $\cos\theta$ |
|---|---|---|
$0°$ | $0$ | $1$ |
$30°$ | $\frac{\pi}{6}$ | $\frac{\sqrt{3}}{2}$ |
$45°$ | $\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2}$ |
$60°$ | $\frac{\pi}{3}$ | $\frac{1}{2}$ |
$90°$ | $\frac{\pi}{2}$ | $0$ |
$120°$ | $\frac{2\pi}{3}$ | $-\frac{1}{2}$ |
$135°$ | $\frac{3\pi}{4}$ | $-\frac{\sqrt{2}}{2}$ |
$150°$ | $\frac{5\pi}{6}$ | $-\frac{\sqrt{3}}{2}$ |
$180°$ | $\pi$ | $-1$ |
$270°$ | $\frac{3\pi}{2}$ | $0$ |
$360°$ | $2\pi$ | $1$ |
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
Evaluate $\cos 180° + \sin 180°$.
Write $\cos 180°$ in radians and state its value.
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.
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