Sin 3 Degrees: Value, Unit Circle & Exact Form

What Is The Value Of Sin 3 Degrees?

The value of sin 3 degrees is approximately $0.0523$, and written more fully $\sin 3^\circ \approx 0.0523359562$. It is a small positive number, which fits an angle that has barely opened from the horizontal.

Here is the same angle in both units, because trigonometry uses both:

Table: Sin 3 degrees expressed in degrees and radians.

Notice that $\sin 3^\circ$ and the angle in radians ($0.0524$) are almost the same number. That is not a coincidence, and the section on why the value is so small explains it. For a wider view of how the three main ratios behave, the sine function page tracks $\sin$ across every angle.

How Do You Find Sin 3 Degrees?

You find sin 3 degrees in three steps that work for any angle: locate the quadrant, find the reference angle, then attach the correct sign.

• Quadrant. $3^\circ$ sits between $0^\circ$ and $90^\circ$, so it is in the first quadrant.

• Reference angle. In the first quadrant the reference angle is the angle itself, so it stays $3^\circ$.

• Sign. Use the ASTC rule (All, Sine, Tangent, Cosine). In the first quadrant all ratios are positive, so $\sin 3^\circ$ is positive.

That settles the sign and size instantly: a first-quadrant sine is positive and, for a small angle, close to zero. The only remaining question is the exact decimal, and for that you need either a table, a calculator, or a series (all covered below).

The right-triangle picture backs this up. Draw a right triangle with one angle of $3^\circ$; then $\sin 3^\circ = \dfrac{\text{opposite}}{\text{hypotenuse}}$. Because the $3^\circ$ corner is so narrow, the opposite side is tiny next to the hypotenuse, so the ratio is small and positive. To review that ratio definition itself, see sin cos tan.

Where Does 3 Degrees Sit On The Unit Circle?

On the unit circle, sin of an angle is the $y$-coordinate of the point where the angle's ray meets the circle. For $3^\circ$ that point is almost due east, barely lifted above the $x$-axis.

$$(\cos 3^\circ,\ \sin 3^\circ) = (0.9986,\ 0.0523)$$

The $x$-coordinate ($\cos 3^\circ = 0.9986$) is nearly $1$, and the $y$-coordinate ($\sin 3^\circ = 0.0523$) is just above $0$. The point has hardly moved from $(1, 0)$, which is exactly what a $3^\circ$ turn should look like.

This double picture, the triangle ratio and the circle height, gives the same $0.0523$. To explore the circle interactively, the unit circle with tangent page adds the tangent line to the same diagram.

Can You Write An Exact Value For Sin 3 Degrees?

Yes, an exact value exists, but it is not simple. This is the honest answer competitors skip.

Angles like $30^\circ$, $45^\circ$, and $60^\circ$ have clean exact sines ($\frac{1}{2}$, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{3}}{2}$). The reason is that they are constructible with compass and straightedge and reduce to short surds. It turns out $3^\circ$ is constructible too, because it can be written as a difference of two constructible angles:

$$3^\circ = 18^\circ - 15^\circ$$

Both $18^\circ$ and $15^\circ$ have known exact sines and cosines, so the sum and difference identities give an exact expression:

$$\sin 3^\circ = \sin(18^\circ - 15^\circ) = \sin 18^\circ \cos 15^\circ - \cos 18^\circ \sin 15^\circ$$

Substituting the known values $\sin 18^\circ = \frac{\sqrt{5}-1}{4}$, $\cos 15^\circ = \frac{\sqrt{6}+\sqrt{2}}{4}$, $\cos 18^\circ = \frac{\sqrt{10+2\sqrt{5}}}{4}$, and $\sin 15^\circ = \frac{\sqrt{6}-\sqrt{2}}{4}$ produces:

$$\sin 3^\circ = \frac{(\sqrt{5}-1)(\sqrt{6}+\sqrt{2}) - \sqrt{10+2\sqrt{5}},(\sqrt{6}-\sqrt{2})}{16}$$

That is a genuine exact value, and it does evaluate to $0.05234$. It is also almost useless for hand calculation, because it hides a nested radical inside a difference of products. So the working value stays the decimal:

$$\sin 3^\circ \approx 0.0523$$

Do not try to force a short answer like $\frac{1}{20}$ or a single simple surd. No such clean form exists for $3^\circ$. The nested-radical expression above, or the decimal, is as good as it gets.

How Does Sin 3 Degrees Compare To Nearby Small Angles?

Small-angle sines climb almost in a straight line at first, roughly matching the angle measured in radians. The table shows the pattern.

Table: Sine of several small angles, with degrees, radians, and the four-decimal value.

The "also equals" column uses the cofunction relation: $\sin\theta = \cos(90^\circ - \theta)$, so $\sin 3^\circ = \cos 87^\circ$. For the individual value pages, see sin 2 degrees, sin 15 degrees, sin 30 degrees, and sin 45 degrees. The full grid lives in the trigonometric table.

Why Is Sin 3 Degrees So Small And Positive?

Two facts explain the value, and both come straight from the definitions above.

• Positive, because of the quadrant. $3^\circ$ is a first-quadrant angle, and by the ASTC rule every ratio is positive there. The unit-circle point sits above the $x$-axis, so its height ($\sin$) is positive.

• Small, because the angle barely opened. On the unit circle the point has moved only $3^\circ$ from $(1, 0)$, so its height above the axis is tiny. In a right triangle the side opposite a $3^\circ$ corner is short next to the hypotenuse.

There is a deeper reason the value $0.0523$ is so close to the radian measure $0.0524$. For small angles measured in radians, $\sin\theta \approx \theta$, called the small-angle approximation. Since $3^\circ = \frac{\pi}{60} \approx 0.05236$ radians and $\sin 3^\circ \approx 0.05234$, the two agree to three decimals, with the sine always a hair smaller than the angle. To see why radian measure makes this work, read what is a radian.

Who Discovered How To Compute Sin 3 Degrees?

Long before calculators, astronomers needed sine values for angles far finer than $30^\circ$ and $45^\circ$, because tracking planets demanded precision. Building tables of small-angle sines was one of the great projects of early mathematics.

Two other figures shaped how small-angle sines are found:

• Ptolemy (c. 100–170 CE, Roman Egypt) built a table of chords in the Almagest, the ancestor of the sine table, with entries in half-degree steps.

• Madhava of Sangamagrama (c. 1340–1425, India) discovered the power series for sine, $\sin x = x - \frac{x^3}{3\cdot 2 \cdot 1} + \cdots$, which is exactly how a modern calculator finds $\sin 3^\circ$ to as many decimals as you want.

Where Is Sin 3 Degrees Used In The Real World?

Tiny angles like $3^\circ$ show up wherever a small tilt or deviation has to be measured precisely.

• Navigation: a ship or aircraft a few degrees off heading drifts far over a long distance, and the cross-track error is computed with the sine of that small angle.

• Surveying and construction: a road camber, a drainage slope, or a wheelchair ramp is often specified as a gentle grade of a few degrees, and its rise is length times $\sin(\text{angle})$.

• Optics and engineering: the small deflection of a laser or a beam is modelled with $\sin\theta$, where $\theta$ is only a couple of degrees.

• Astronomy: the apparent shift of a nearby star (parallax) is a minute angle, and its sine converts the angle into a real distance.

One small ratio quietly does a lot of work. Precision at $3^\circ$ is the difference between a ship reaching port and missing it by miles.

What Are The Most Common Mistakes With Sin 3 Degrees?

These four errors account for most wrong answers on small-angle sines, and each has a clean fix.

Leaving the calculator in radian mode.

Where it slips in:

A student types sin(3) expecting $\sin 3^\circ$, but the calculator is set to radians and returns $0.1411$, the sine of $3$ radians (about $172^\circ$).

Don't do this:

Do not trust the display until you have checked the angle mode.

The correct way:

Set the calculator to degree mode before entering $3$, or convert first: $3^\circ = \frac{\pi}{60}$ radians, then take the sine. The right answer is $0.0523$, not $0.1411$.

Reading 3 degrees as 3 radians.

Where it slips in:

A student writes $\sin 3^\circ = \sin 3$ and computes with the bare number $3$, mixing the two angle units.

Don't do this:

Do not drop the degree symbol or assume the number alone carries the unit.

The correct way:

Keep the unit attached. $3^\circ$ is a small angle near the start of the circle; $3$ radians is a large angle past a right angle. They give very different sines.

Inventing a simple exact value.

Where it slips in:

A student assumes every angle has a neat surd like $\frac{\sqrt{3}}{2}$ and writes a made-up short form for $\sin 3^\circ$.

Don't do this:

Do not force a clean radical. Only special angles reduce to short surds; $3^\circ$ only has a messy nested-radical form.

The correct way:

Use the decimal $0.0523$ for calculation, and quote the $18^\circ - 15^\circ$ nested-radical expression only when an exact form is explicitly required.

Confusing the cofunction relation.

Where it slips in:

A student remembers a $\sin$-$\cos$ link but writes $\sin 3^\circ = \cos 3^\circ$ instead of $\cos 87^\circ$.

Don't do this:

Do not pair the same angle. The cofunction swaps the angle for its complement.

The correct way:

Use $\sin\theta = \cos(90^\circ - \theta)$, so $\sin 3^\circ = \cos 87^\circ = 0.0523$. The cofunction identities page lists the full set.

Practice Problems On Sin 3 Degrees

Try each, then check the answer that follows.

1. Convert $3^\circ$ to radians.
   (Answer: $3^\circ \times \frac{\pi}{180} = \frac{\pi}{60} \approx 0.0524$ rad.)

2. Using the cofunction relation, write $\sin 3^\circ$ as a cosine.
   (Answer: $\sin 3^\circ = \cos 87^\circ = 0.0523$.)

3. A ramp is $12$ m long and rises at $3^\circ$. How high is the top?
   (Answer: height $= 12 \sin 3^\circ = 12 \times 0.0523 \approx 0.628$ m.)

4. Round $\sin 3^\circ = 0.0523359562$ to three decimal places.
   (Answer: $0.052$.)

5. Is $\sin 3^\circ$ larger or smaller than $\frac{\pi}{60}$?
   (Answer: smaller. $\sin 3^\circ \approx 0.05234$ and $\frac{\pi}{60} \approx 0.05236$, since $\sin\theta < \theta$ for small positive angles.)

6. Which is bigger, $\sin 3^\circ$ or $\sin 5^\circ$?
   (Answer: $\sin 5^\circ = 0.0872$ is bigger, because sine increases across the first quadrant.)

Where Should You Go Next After Sin 3 Degrees?

Sin 3 degrees is one entry in a much larger pattern, and a few natural doors open from here.

1. Trigonometric table. See every standard angle's sine, cosine, and tangent in one grid, so $3^\circ$ sits in context.

2. Sum and difference identities. The tool that produced the exact $18^\circ - 15^\circ$ form, useful for building many other values.

3. What is a radian. Understand the unit that makes the small-angle approximation work.

If your child is building these foundations, a live Bhanzu trainer teaches trigonometry starting from the "why" (the circle and the triangle behind every value) in the Bhanzu trigonometry program.