What Is The Value Of Sin 270 Degrees?
Sin 270 Degrees is exactly $-1$. Written with the angle in radians, $\sin\frac{3\pi}{2} = -1$, since $270^\circ$ and $\frac{3\pi}{2}$ radians name the same angle. As a decimal the value is $-1.0000$, and it is exact, not rounded.
The angle $270^\circ$ is a quadrantal angle: its terminal side lands directly on an axis instead of pointing into one of the four quadrants. Starting from the positive x-axis and turning anticlockwise, $90^\circ$ points straight up, $180^\circ$ points left, and $270^\circ$ points straight down along the negative y-axis.
On the unit circle, the point at $270^\circ$ is $(0, -1)$. Sine always equals the y-coordinate of that point, so $\sin 270^\circ = -1$. Cosine equals the x-coordinate, which is $0$ here, and that single fact separates the two functions at this angle.
How Do You Find Sin 270 Degrees?
To find sine at any angle, locate where the angle's terminal side meets the unit circle and read the y-coordinate. For $270^\circ$ the terminal side runs straight down, so it meets the circle at $(0, -1)$, and the y-coordinate is $-1$.
Most special angles ask you to build a reference angle, the acute angle between the terminal side and the x-axis, then attach a sign from the quadrant. That method needs a triangle. A quadrantal angle has no such triangle, because a terminal side lying flat on an axis cannot form a right triangle with that same axis.
So the honest answer is that the sign rule you may have memorised, often taught as ASTC (or CAST), is not what settles $270^\circ$. The point itself does. Here is the short version, in both angle systems:
In degrees: the terminal side of $270^\circ$ meets the unit circle at $(0, -1)$, so $\sin 270^\circ = -1$.
In radians: the terminal side of $\frac{3\pi}{2}$ meets the unit circle at the same point, so $\sin\frac{3\pi}{2} = -1$.
Where Does 270 Degrees Sit On The Unit Circle?
The angle $270^\circ$ sits at the bottom of the unit circle, three-quarters of the way around from the positive x-axis. Its coordinates are $(0, -1)$: zero across, one unit down.
The unit circle also makes the neighbours easy to compare. At $180^\circ$ the point is $(-1, 0)$, so sin 180 degrees is $0$. At $90^\circ$ the point is $(0, 1)$, so sin 90 degrees is $+1$. At $270^\circ$ the point flips to $(0, -1)$, the mirror image straight down, so the sine flips to $-1$.
For the radian-first view of this exact angle, see sin 3pi 2, and for the tangent line drawn on the same diagram, see the unit circle with tangent.
Can You Find Sin 270 Degrees Without The Unit Circle?
Yes. You can split $270^\circ$ into two angles you already know and apply the sine angle-sum formula, $\sin(A + B) = \sin A \cos B + \cos A \sin B$. This gives the same exact $-1$ with no picture at all.
Write $270^\circ$ as $180^\circ + 90^\circ$:
$$\sin 270^\circ = \sin(180^\circ + 90^\circ)$$
$$\sin 270^\circ = \sin 180^\circ \cos 90^\circ + \cos 180^\circ \sin 90^\circ$$
$$\sin 270^\circ = (0)(0) + (-1)(1)$$
$$\sin 270^\circ = -1$$
A second route uses the co-function relation for angles measured back from $270^\circ$: $\sin(270^\circ - \theta) = -\cos\theta$. Setting $\theta = 0^\circ$ gives $\sin 270^\circ = -\cos 0^\circ = -(1) = -1$. Both methods agree with the point $(0, -1)$.
What Are The Values Around Sin 270 Degrees?
The quadrantal angles share one feature: each sits on an axis, so each sine and cosine is either $0$, $+1$, or $-1$. The table sets $270^\circ$ beside its family, with degrees and radians together.
Table: Sine, cosine, and tangent at the quadrantal angles, in degrees and radians.
Angle | Radians | sin | cos | tan |
|---|---|---|---|---|
$0^\circ$ | $0$ | $0$ | $1$ | $0$ |
$90^\circ$ | $\frac{\pi}{2}$ | $1$ | $0$ | undefined |
$180^\circ$ | $\pi$ | $0$ | $-1$ | $0$ |
$270^\circ$ | $\frac{3\pi}{2}$ | $-1$ | $0$ | undefined |
$360^\circ$ | $2\pi$ | $0$ | $1$ | $0$ |
Tangent is $\frac{\sin}{\cos}$, so at $270^\circ$ it divides $-1$ by $0$ and is undefined. For the full set of standard angles between these, the trigonometric table and the page on trigonometric ratios of specific angles lay them out in one place.
Why Is Sin 270 Degrees Equal To -1?
Sin 270 Degrees is $-1$ because the angle points straight down, and straight down on the unit circle is exactly one unit below the centre. The reasoning is short when you keep it tied to the circle.
The point decides the value. Sine is defined as the y-coordinate of the point where the terminal side meets the unit circle. At $270^\circ$ that point is $(0, -1)$, so the sine is the y-value, $-1$.
Below the axis means negative. Any angle whose terminal point sits below the x-axis has a negative y-coordinate, hence a negative sine. The bottom of the circle is the most negative it can get, exactly $-1$.
It is the mirror of $90^\circ$. At $90^\circ$ the point is $(0, +1)$ and sine is $+1$. Reflecting straight down to $270^\circ$ flips the y-coordinate to $-1$, so the sine flips sign but keeps the size.
That last point is worth holding onto. The size of the value, $1$, is the radius of the circle, and the sign, negative, is only the direction. Sine at a quadrantal angle is never a messy decimal, only $0$, $+1$, or $-1$.
Who Discovered The Sine Function?
Sine did not begin in Europe, and it did not begin as a ratio in a triangle. It began in India as a table of half-chords, built by astronomers who needed to predict the positions of the sun and planets.
Two other figures shaped the same idea:
Hipparchus of Nicaea (c. 190 to 120 BCE, Greece) is often called the founder of trigonometry for building the first known table of chords, the Greek ancestor of the sine table.
Claudius Ptolemy (c. 100 to 170 CE, Roman Egypt) extended those chords in the Almagest, the reference astronomers used for more than a thousand years.
Where Is Sin 270 Degrees Used In The Real World?
The value $-1$ marks a trough, the lowest point of a smooth back-and-forth motion, and that shows up wherever something oscillates or turns in a circle.
Alternating current: household electricity rises and falls as a sine wave, and the instant the wave reaches its most negative peak corresponds to the $270^\circ$ point of the cycle.
Sound and music: a pure tone is a sine wave, and its trough, where the air pressure is lowest, sits at the same three-quarter mark of each vibration.
Circular motion and rides: a point on a turning Ferris wheel or a spinning gear reaches its lowest position at $270^\circ$ of the turn, exactly where the vertical coordinate hits $-1$ times the radius.
Computer graphics: rotating a sprite or a 3D model uses sine and cosine of the turn angle, and a quarter-turn past the halfway mark plugs in $\sin 270^\circ = -1$ directly.
Navigation and GPS: position models built on circular and elliptical motion evaluate sine at every angle of the orbit, including the straight-down $270^\circ$ mark.
One value, $-1$, quietly labels the bottom of every wave and every turn around you.
What Are The Most Common Mistakes With Sin 270 Degrees?
Three errors account for most of the wrong answers on quadrantal angles. Each one is easy to avoid once you name it.
Hunting for a reference-angle triangle.
Where it slips in:
A student trained on $30^\circ$, $45^\circ$, and $60^\circ$ tries to build a reference triangle for $270^\circ$ and force it into the third quadrant.
Don't do this:
Do not draw a triangle for a quadrantal angle. The terminal side lies flat on the y-axis, so no right triangle exists between the side and that axis.
The correct way:
Read the coordinates of the point on the unit circle. At $270^\circ$ the point is $(0, -1)$, and the sine is the y-coordinate, $-1$, with no triangle needed.
Leaving the calculator in the wrong mode.
Where it slips in:
A student types $270$ expecting $-1$, but the calculator is set to radians, so it evaluates $\sin(270\text{ rad})$ and returns roughly $-0.176$.
Don't do this:
Do not trust the display before checking the angle mode. The number $270$ means very different angles in degrees and in radians.
The correct way:
Set the calculator to degrees for $\sin 270^\circ$, or enter $\frac{3\pi}{2}$ when the mode is radians. Both give exactly $-1$.
Swapping sine and cosine at 270 degrees.
Where it slips in:
A student writes $\sin 270^\circ = 0$ and $\cos 270^\circ = -1$, exchanging the two coordinates of the point.
Don't do this:
Do not confuse which coordinate belongs to which function. Sine is the y-coordinate; cosine is the x-coordinate.
The correct way:
At the point $(0, -1)$, the y-coordinate $-1$ is $\sin 270^\circ$ and the x-coordinate $0$ is $\cos 270^\circ$. Keep y with sine and x with cosine.
Practice Problems On Sin 270 Degrees
Work each one from the unit circle or the quadrantal values, then check against the answer.
Evaluate $\sin 270^\circ + \cos 180^\circ$.
(Answer: $-1 + (-1) = -2$.)Write $\sin\frac{3\pi}{2}$ as a single value.
(Answer: $-1$.)Simplify $2\sin 270^\circ - \sin 90^\circ$.
(Answer: $2(-1) - 1 = -3$.)Find $\sin 270^\circ \times \cos 270^\circ$.
(Answer: $(-1)(0) = 0$.)Is $\sin 270^\circ$ positive or negative, and why?
(Answer: negative, because the terminal point $(0, -1)$ has a negative y-coordinate.)Convert $270^\circ$ to radians and give its sine.
(Answer: $270^\circ = \frac{3\pi}{2}$, and $\sin\frac{3\pi}{2} = -1$.)
Where Should You Go Next After Sin 270 Degrees?
Sin 270 Degrees is one anchor point on the unit circle, and a few natural doors open from here.
Sin 3pi 2. The same angle written in radians, useful once you move from degrees into calculus-style work.
Trigonometric ratios of specific angles. Fill in the whole set of standard angles so the quadrantal values sit inside a bigger map.
Trigonometric table. A single reference for sine, cosine, and tangent across the common angles.
If your child is building the unit circle from the ground up, a live Bhanzu trainer teaches these values starting from the point on the circle, not from memorised tables, in the Bhanzu trigonometry program.
Was this article helpful?
Your feedback helps us write better content
