Cos 3pi/2 - Value 0, Unit Circle, and How to Find It

What Does Cos 3pi/2 Mean?

Cosine is one of the three core trigonometric ratios, and on the unit circle it is the $x$-coordinate of the point where the angle's radius meets the circle. So $\cos \dfrac{3\pi}{2}$ asks: after rotating $\dfrac{3\pi}{2}$ radians from the positive $x$-axis, what is the $x$-coordinate?

A radian measure of $\dfrac{3\pi}{2}$ is three-quarters of a full turn. That rotation ends on the negative $y$-axis at $(0, -1)$, and since the point sits on the $y$-axis, its horizontal position is $0$ - so the cosine is $0$.

Where Does Cos 3pi/2 Show Up?

The angle $\dfrac{3\pi}{2}$ marks the three-quarter point of a full rotation, so it appears wherever a cycle is tracked past its halfway mark. In a rotating wheel, an AC voltage waveform, or a pendulum's swing, $\dfrac{3\pi}{2}$ is the phase where the horizontal component has dropped back to zero and the vertical component is at its most negative.

A cosine of $0$ is the signature of "pointing straight down or straight up" - no horizontal reach at all. That is why the cosine function crosses zero at every quadrantal angle on the vertical axis, a rhythm that repeats through every cycle of periodic motion described by trigonometric functions.

Standard-Angle Reference Table

The angle $\dfrac{3\pi}{2}$ is a quadrantal angle, meaning it sits exactly on an axis. Reading it against the other quadrantal angles makes the value obvious.

Cosine reads the $x$-coordinate. At both $\dfrac{\pi}{2}$ and $\dfrac{3\pi}{2}$ the point sits on the vertical axis, where $x = 0$, so both cosines are $0$.

How Do You Find The Exact Value Of Cos 3pi/2?

There are two clean routes, and both give $0$.

Method 1: The unit circle.

Rotate the radius $\dfrac{3\pi}{2}$ radians, which is three quarter-turns anticlockwise. The tip lands at the bottom of the circle, on the negative $y$-axis.

$$\cos \frac{3\pi}{2} = x\text{-coordinate at }(0, -1) = 0$$

Because the point falls in neither the left nor right half - it is exactly on the axis - the horizontal distance is zero. This sits inside the quadrant framework as a boundary angle between Quadrant III and Quadrant IV.

Method 2: Convert to degrees.

Change the angle first: $\dfrac{3\pi}{2}$ radians $= \dfrac{3}{2} \times 180^\circ = 270^\circ$. The cosine at $270^\circ$ reads the horizontal position at the bottom of the circle.

$$\cos \frac{3\pi}{2} = \cos 270^\circ = 0$$

Examples Of Cos 3pi/2

Example 1

Evaluate $7\cos \dfrac{3\pi}{2}$.

$$7\cos \frac{3\pi}{2} = 7 \times 0 = 0$$

Example 2

Simplify $\cos \dfrac{3\pi}{2} + \sin \dfrac{3\pi}{2}$.

Wrong attempt. A student remembers that $\dfrac{3\pi}{2}$ is "near $\pi$" and writes $\cos \dfrac{3\pi}{2} = -1$, copying the value from $\cos \pi$.

That breaks on the circle: $\pi$ sits at $(-1, 0)$ but $\dfrac{3\pi}{2}$ sits at $(0, -1)$, a different point entirely. The $x$-coordinate at $(0, -1)$ is $0$, not $-1$.

Correct. $\cos \dfrac{3\pi}{2} = 0$ and $\sin \dfrac{3\pi}{2} = -1$ (the $y$-coordinate), so the sum is $0 + (-1) = -1$.

Example 3

Find $\cos \dfrac{3\pi}{2} \times \tan \dfrac{3\pi}{2}$ where it is defined, and explain the result.

Since $\cos \dfrac{3\pi}{2} = 0$, the product with any finite factor is $0$. Note that $\tan \dfrac{3\pi}{2}$ is undefined because it divides by $\cos \dfrac{3\pi}{2} = 0$, so the expression is only meaningful as the limit toward $0$ from the cosine factor.

$$\cos \frac{3\pi}{2} = 0 \implies 0 \times (\text{anything finite}) = 0$$

Example 4

Verify the identity $\cos^2 \dfrac{3\pi}{2} + \sin^2 \dfrac{3\pi}{2} = 1$.

$$(0)^2 + (-1)^2 = 0 + 1 = 1$$

The Pythagorean identity holds at the quadrantal angle exactly as it does everywhere else.

Example 5

A crank arm of length $0.5$ m has turned through $\dfrac{3\pi}{2}$ radians. Find its horizontal distance from the pivot.

The horizontal distance is $0.5\cos \dfrac{3\pi}{2}$.

$$0.5 \times \cos \frac{3\pi}{2} = 0.5 \times 0 = 0 \text{ m}$$

The arm points straight down, so it has no horizontal reach.

Where Students Trip Up On Cos 3pi/2

Mistake 1: Swapping the sine and cosine at 270 degrees

Where it slips in: Recall of quadrantal values, where $0$ and $-1$ both belong to $\dfrac{3\pi}{2}$ but attach to different functions.

Don't do this: Writing $\cos \dfrac{3\pi}{2} = -1$. That is the sine at $\dfrac{3\pi}{2}$; the cosine is $0$.

The correct way: Cosine is the $x$-coordinate and sine is the $y$-coordinate. At $(0, -1)$ the $x$ is $0$ (cosine) and the $y$ is $-1$ (sine). Students who memorise the pair without tagging which coordinate is which reliably reverse them here.

Mistake 2: Confusing cos 3pi/2 with cos 3pi

Where it slips in: Skim-reading the angle, where $\dfrac{3\pi}{2}$ and $3\pi$ look similar.

Don't do this: Reporting $\cos \dfrac{3\pi}{2} = -1$, which is actually the value of cos 3pi at the $(-1, 0)$ position.

The correct way: $\dfrac{3\pi}{2}$ is $270^\circ$ and lands at $(0, -1)$, so its cosine is $0$; $3\pi$ is $540^\circ$ and lands at $(-1, 0)$, so its cosine is $-1$. The denominator changes everything.

Mistake 3: Treating a quadrantal angle like a right triangle

Where it slips in: Applying SOH-CAH-TOA at $270^\circ$, where no genuine right triangle exists.

Don't do this: Trying to build an adjacent-over-hypotenuse ratio at $\dfrac{3\pi}{2}$ and getting stuck on a "triangle" with zero width.

The correct way: For quadrantal angles, read the coordinate straight off the unit circle. The reference angle method is for angles strictly inside a quadrant; on an axis, the point already gives the answer.

Key Takeaways

• Cos 3pi/2 equals $0$, an exact quadrantal value, because the angle lands at $(0, -1)$ on the unit circle.

• Cosine reads the $x$-coordinate; at $\dfrac{3\pi}{2}$ that coordinate is $0$, while sine there is $-1$.

• Read quadrantal angles straight off the circle rather than forcing a right triangle.

• To take $\cos \dfrac{3\pi}{2}$ further with a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or live math classes online.

Practice These Before Moving On

1. Evaluate $4\cos \dfrac{3\pi}{2} - 3\sin \dfrac{3\pi}{2}$.

2. Show that $\cos \dfrac{3\pi}{2} = \cos \dfrac{7\pi}{2}$ using periodicity.

3. A wheel spoke of length $0.3$ m has rotated $\dfrac{3\pi}{2}$ radians. Find its horizontal displacement from centre.

Want a live Bhanzu trainer to walk through more cos 3pi/2 problems? Book a free demo class.

Read More

• Cos 270 Degrees — the same value read from the degree measure of this angle.

• Cos 180 Degrees — the quadrantal value at the $(-1, 0)$ position.

• Cos 2pi — the cosine after a full turn back to $(1, 0)$.

• Trigonometric Table — sine, cosine, and tangent for every standard angle in one chart.