Sin 3pi/2 : Exact Value -1 and How to Find I

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.

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

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

2. Sketch one cycle of $y = \sin\theta$ and mark the point at $\theta = \dfrac{3\pi}{2}$.

3. 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.