Quick Answer:
Result: $\cos 270° = 0$
Notation: $\cos\left(\dfrac{3\pi}{2}\right) = 0$ (radians)
Method shown: unit circle — the $x$-coordinate at three-quarters of a turn
Approximate value: $0$ (exact)
Exact form: $0$
A rotation of $270°$ — three-quarters of the way around the unit circle, or $\frac{3\pi}{2}$ radians — lands on the point at the very bottom, $(0, -1)$. Cosine reads the $x$-coordinate of where you land, and at the bottom of the circle that $x$-coordinate is $0$. The point sits directly below the center, so it has no horizontal offset at all.
Quick Reference Table — Cosine of Standard Angles
This table lists cosine at the standard angles in both degrees and radians, with $270°$ highlighted.
Angle (degrees) | Angle (radians) | $\cos\theta$ |
|---|---|---|
$0°$ | $0$ | $1$ |
$90°$ | $\frac{\pi}{2}$ | $0$ |
$180°$ | $\pi$ | $-1$ |
$210°$ | $\frac{7\pi}{6}$ | $-\frac{\sqrt{3}}{2}$ |
$225°$ | $\frac{5\pi}{4}$ | $-\frac{\sqrt{2}}{2}$ |
$240°$ | $\frac{4\pi}{3}$ | $-\frac{1}{2}$ |
$270°$ | $\frac{3\pi}{2}$ | $0$ |
$300°$ | $\frac{5\pi}{3}$ | $\frac{1}{2}$ |
$315°$ | $\frac{7\pi}{4}$ | $\frac{\sqrt{2}}{2}$ |
$330°$ | $\frac{11\pi}{6}$ | $\frac{\sqrt{3}}{2}$ |
$360°$ | $2\pi$ | $1$ |
Cosine crosses zero twice per turn — at $90°$ and again at $270°$. Both are points where the circle is at its top or bottom, directly above or below the center, with no horizontal distance from the $y$-axis.
Where cos 270 Degrees Appears
The value $\cos 270° = 0$ marks a quarter-cycle turning point in any oscillation. In a cosine wave modelling a swinging pendulum or an AC voltage, the moment the phase reaches $270°$ is when the horizontal cosine component is exactly zero and the motion is passing through its midline.
The same zero defines the vertical axis in screen and robotics geometry: a heading of $270°$ points straight down, with no left-right component, which is why navigation and robotics control systems treat $270°$ as a pure-vertical bearing. Wherever a rotating quantity has "no horizontal part," $\cos 270° = 0$ is the number saying so.
What is cos 270 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. A turn of $270°$ covers three of the four quarter-turns.
That rotation lands on $(0, -1)$ — the lowest point of the circle. The $x$-coordinate there is $0$, so $\cos 270° = 0$. Written in radians the angle is $\frac{3\pi}{2}$, which is why $\cos 270°$ and $\cos\left(\frac{3\pi}{2}\right)$ are the same number.
How To Find The Value of cos 270 Degrees
Method 1 — Unit circle
Rotate $270°$ counterclockwise from $(1, 0)$. Three quarter-turns put you at the bottom of the circle, the point $(0, -1)$.
Cosine is the $x$-coordinate of that point.
Final answer: $\cos 270° = 0$.
Method 2 — Cosine subtraction formula
Write $270°$ as $360° - 90°$ and apply $\cos(A - B) = \cos A\cos B + \sin A\sin B$:
$$\cos(360° - 90°) = \cos 360°\cos 90° + \sin 360°\sin 90°$$
$$= (1)(0) + (0)(1) = 0$$
Final answer: $\cos 270° = 0$.
Method 3 — Co-function shift
Use $\cos(270°) = \cos(180° + 90°) = -\cos 90°$ via the periodic shift, and $\cos 90° = 0$:
$$\cos 270° = -\cos 90° = -(0) = 0$$
Zero has no sign, so the negative in front changes nothing here.
Common mistakes with cos 270 degrees
Mistake 1: Confusing cos 270° with sin 270°
Where it slips in: At $270°$ the unit-circle point is $(0, -1)$, and the two coordinates get swapped.
Don't do this: Writing $\cos 270° = -1$ (that is $\sin 270°$, the $y$-coordinate).
The correct way: Cosine is the $x$-coordinate, so $\cos 270° = 0$; sine is the $y$-coordinate, so $\sin 270° = -1$.
This swap is the recurring trap on quadrantal angles — a Mathway unit-circle walkthrough for $\cos 270°$ exists precisely because so many learners reach for $-1$.
Mistake 2: Calculator in degree vs radian mode
Where it slips in: Typing cos(270) while the calculator is set to radians.
Don't do this: Reading $\cos(270 \text{ rad}) \approx 0.984$ and reporting it as $\cos 270°$.
The correct way: Use degree mode for $\cos 270°$, or enter $\cos\left(\frac{3\pi}{2}\right)$ in radian mode. Both give $0$.
Mistake 3: Writing 0 with a sign
Where it slips in: Carrying the second-quadrant intuition that "cosine is negative here" into the answer.
Don't do this: Writing $\cos 270° = -0$ or treating the result as a small negative number.
The correct way: The result is exactly $0$ — zero has no sign, so $\cos 270° = 0$, plain.
Conclusion
Cos 270 degrees equals $0$ — the cosine of three-quarters of a turn, written $\cos\left(\frac{3\pi}{2}\right)$ in radians.
The rotation lands on $(0, -1)$, the bottom of the unit circle, where the $x$-coordinate is $0$.
Three routes agree: unit circle, the subtraction formula on $360° - 90°$, and the co-function shift.
The most common slip is swapping cosine and sine — $\cos 270° = 0$, $\sin 270° = -1$.
Quick self-check — try these
Evaluate $\cos 270° + \sin 270°$.
Write $\cos 270°$ in radians and state its value.
Explain in one sentence why $\cos 270°$ and $\cos 90°$ are equal.
If #1 didn't give $-1$, recheck the coordinates of the point at $270°$. Want a live Bhanzu trainer to walk through more unit-circle problems? Book a free demo class — online globally.
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