Cos 270 Degrees = 0 — Value, Unit Circle, Radians
Cos 270 Degrees = 0 — Value, Unit Circle, Radians
TL;DR
The value of cos 270 degrees is exactly 0. In radians, 270° is (\frac{3\pi}{2}), so (\cos(270°) = \cos\left(\frac{3\pi}{2}\right) = 0). This article shows why three-quarters of a turn lands on the bottom of the unit circle where the x-coordinate vanishes, provides a standard-angle reference table in degrees and radians, and clarifies common mix-ups.
Quick Answer:
- Result: (\cos 270° = 0)
- Notation: (\cos(3\pi/2) = 0) (radians)
- Method shown: unit circle — the x-coordinate at three-quarters of a turn
- Approximate value: 0 (exact)
A rotation of 270°
A rotation of 270° — three-quarters of the way around the unit circle, or (\frac{3\pi}{2}) radians — lands at the point (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
| Angle (degrees) | Angle (radians) | (\cos θ) |
|---|---|---|
| 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 modeling a swinging pendulum or an AC voltage, the object is at midline when the phase reaches 270°. The same zero defines the vertical axis in screen and robotics geometry: a heading of 270° points straight down, signifying no left-right component, which is why navigation and robotics treat it as a vertical bearing.
What is cos 270 Degrees?
Cosine of an angle is 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. Its x-coordinate there is 0, so (\cos 270° = 0).
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, at (0,−1).
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°)).
Final answer: (\cos 270° = 0).
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). Don't confuse the coordinates.
Correct: (\cos 270° = 0 \sin 270° = -1).
Mistake 2: Calculator in Degree vs Radian Mode
Reading (\cos(270)) while in radians can lead to incorrect results.
Correct Approach: Use degree mode for (\cos 270°) or enter (\cos(3\pi/2)) in radian mode.
Mistake 3: Writing 0 with a Sign
Zero has no sign. Writing (\cos 270° = -0) is incorrect.
Correct: (\cos 270° = 0).
Conclusion
- Cos 270 degrees equals 0 — the cosine of three-quarters of a turn, written (\cos(3\pi/2)) in radians.
- The rotation lands on (0,−1), the bottom of the unit circle, where the x-coordinate is 0.
- Three methods support this: unit circle, subtraction formula, and co-function shift.
- A common slip is confusing 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.
Frequently Asked Questions
What is the value of cos 270 degrees? 0.
What is the exact value of cos 270 degrees? Exactly 0.
What is cos 270 degrees in radians? 270° equals (\frac{3\pi}{2}) radians.
Is cos 270° the same as cos 90°? Yes, both are 0.
Is cos 270° positive or negative? Neither — it is exactly 0.