What Does Sin 3pi/2 Mean?
Sine is one of the core trigonometric ratios in radians, and on the unit circle it is defined as the $y$-coordinate of the point where the angle's radius meets the circle. A quadrantal angle is one whose terminal side falls on an axis rather than inside a quadrant.
So $\sin\dfrac{3\pi}{2}$ asks: after rotating three-quarters of a full turn, how high above the horizontal axis does the point sit? It sits below the axis, at height $-1$, the deepest sine can go.
The negative sign is not a rounding artefact, it records that the point is below the $x$-axis. Sine is negative for every angle whose terminal side dips beneath the horizontal.
Where Does Sin 3pi/2 Show Up?
Anything that swings between a high and a low (a spring, a tide, an alternating current written as $\sin\theta$) hits its lowest value when the phase reaches $\dfrac{3\pi}{2}$. That is the trough of the wave, the mirror of the peak at $\dfrac{\pi}{2}$.
The value also anchors the four "corners" of the unit circle. Knowing that $\sin\dfrac{3\pi}{2} = -1$ is what lets you sketch a sine graph from memory, because the trough is nailed down.
Standard-Angle Reference Table
$\dfrac{3\pi}{2}$ is a quadrantal angle, it lands exactly on an axis, not inside a quadrant. Here are the sine values at the four quadrantal angles, in radians and degrees.
Angle (radians) | Angle (degrees) | Terminal point | $\sin\theta$ |
|---|---|---|---|
$0$ | $0^\circ$ | $(1, 0)$ | $0$ |
$\dfrac{\pi}{2}$ | $90^\circ$ | $(0, 1)$ | $1$ |
$\pi$ | $180^\circ$ | $(-1, 0)$ | $0$ |
$\dfrac{3\pi}{2}$ | $270^\circ$ | $(0, -1)$ | $-1$ |
Read the sine column and it swings $0 \to 1 \to 0 \to -1$ across the four quarter-turns. The minimum, $-1$, lands at $\dfrac{3\pi}{2}$.
How Do You Find The Value Of Sin 3pi/2?
Two routes lead to $-1$.
Method 1: Read the unit circle.
Rotate the radius counter-clockwise through three quarter-turns: past $\dfrac{\pi}{2}$ (up), past $\pi$ (left), to $\dfrac{3\pi}{2}$ (straight down). The tip lands at $(0, -1)$.
$$\sin\frac{3\pi}{2} = y\text{-coordinate} = -1$$
Method 2: Convert to degrees.
Multiply by $\dfrac{180^\circ}{\pi}$:
$$\frac{3\pi}{2} \times \frac{180^\circ}{\pi} = 270^\circ$$
At $270^\circ$ the radius points straight down, so $\sin 270^\circ = -1$. The degree form and the radian form name the same angle, so $\sin\dfrac{3\pi}{2} = \sin 270^\circ = -1$.
Because $270^\circ$ sits on the negative $y$-axis, no reference-angle triangle is needed here, the value is read directly off the axis.
Examples Of Sin 3pi/2
Example 1
Evaluate $4\sin\dfrac{3\pi}{2}$.
$$4\sin\frac{3\pi}{2} = 4 \times (-1) = -4$$
Example 2
Find $\sin\dfrac{3\pi}{2}$. A student reasons that since $\dfrac{3\pi}{2}$ is bigger than $\pi$, the sine "must be positive like sin at the start".
Wrong attempt. Assuming sine stays positive because the angle is large.
That breaks at the axis test: $\dfrac{3\pi}{2}$ points downward to $(0, -1)$, and any point below the $x$-axis has a negative $y$-coordinate. So a positive answer contradicts the picture.
Correct. Sine is the $y$-coordinate, and at $(0, -1)$ that coordinate is $-1$. The size of the angle does not fix the sign; the position on the circle does.
Example 3
Evaluate $\sin\dfrac{3\pi}{2} + \cos\dfrac{3\pi}{2}$.
$$\sin\frac{3\pi}{2} = -1, \qquad \cos\frac{3\pi}{2} = 0$$ $$\sin\frac{3\pi}{2} + \cos\frac{3\pi}{2} = -1 + 0 = -1$$
Example 4
Verify $\sin^2\dfrac{3\pi}{2} + \cos^2\dfrac{3\pi}{2} = 1$.
$$(-1)^2 + (0)^2 = 1 + 0 = 1$$
The Pythagorean identity checks out.
Example 5
A mass on a spring has vertical position $y = \sin\theta$. Find its position at $\theta = \dfrac{3\pi}{2}$ and describe it.
$$y = \sin\frac{3\pi}{2} = -1$$
The mass is at its lowest point, one unit below the rest line, the trough of its motion.
Where Students Trip Up On Sin 3pi/2
Mistake 1: Dropping the negative sign
Where it slips in: Recalling the size $1$ but forgetting the angle points downward.
Don't do this: Writing $\sin\dfrac{3\pi}{2} = 1$. That is $\sin\dfrac{\pi}{2}$, the peak, the opposite end of the wave.
The correct way: The point $(0, -1)$ sits below the axis, so the $y$-coordinate is $-1$. The memorizer who stores "sine of a quarter-ish angle is $1$" without the sign is the one who loses the mark here.
Mistake 2: Reading the x-coordinate instead of the y
Where it slips in: At the axis points, students grab whichever coordinate is nonzero.
Don't do this: Writing $\sin\dfrac{3\pi}{2} = 0$ by copying the $x$-coordinate.
The correct way: Sine is always the $y$-coordinate. At $(0, -1)$ that is $-1$; the $0$ is the $x$-coordinate and belongs to cosine.
Mistake 3: Converting 3π/2 to the wrong degree measure
Where it slips in: Rushing the radian-to-degree step and landing on $135^\circ$ or $180^\circ$.
Don't do this: Writing $\dfrac{3\pi}{2} = 135^\circ$ (that is $\dfrac{3\pi}{4}$).
The correct way: Multiply the whole angle by $\dfrac{180^\circ}{\pi}$: $\dfrac{3\pi}{2}\times\dfrac{180^\circ}{\pi} = 270^\circ$. Convert first, then read the circle.
Key Takeaways
Sin 3pi/2 equals $-1$, the minimum value the sine function reaches.
The angle $\dfrac{3\pi}{2}$ is $270^\circ$; it points straight down to $(0, -1)$, and sine reads that $y$-coordinate.
The most common slip is dropping the negative sign and writing $1$, that is $\sin\dfrac{\pi}{2}$, the peak, not the trough.
As a quadrantal angle it needs no reference triangle, read it off the axis.
To build fluency with the whole unit circle alongside a teacher, explore Bhanzu's trigonometry tutor or a high school math tutor. The degree form of this angle is covered at sin 270°.
Practice These Before Moving On
Evaluate $3\sin\dfrac{3\pi}{2} + 2\cos\dfrac{3\pi}{2}$.
Sketch one cycle of $y = \sin\theta$ and mark the point at $\theta = \dfrac{3\pi}{2}$.
An AC signal is $v = \sin\theta$. Find its value at $\theta = \dfrac{3\pi}{2}$ and state whether it is a peak or a trough.
Want a live Bhanzu trainer to walk through more quadrantal-angle problems? Book a free demo class, online with an expert, anywhere.
For the formal picture of the four axis points, see the reference on the unit circle.
Read More
Cos 270 degrees, the cosine at the same point on the circle.
Tan 270 degrees, the tangent at $270^\circ$ and why it is undefined.
Sin 3π, another large radian angle reduced on the circle.
270 degree angle, the angle itself, from a geometry view.
Trigonometric table, sine, cosine, tangent at the standard angles.
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