Sin 4 Degrees: Value, Unit Circle & How To Find It

What Is The Value Of Sin 4 Degrees?

Sin 4 degrees is approximately $0.0698$, or more precisely $\sin 4^\circ = 0.069756$ to six decimal places. It is a small positive value. Written both ways an angle should always be written, the input is $4^\circ$ in degrees and $\frac{\pi}{45} \approx 0.069813$ in radians.

$$\sin 4^\circ = \sin\frac{\pi}{45} \approx 0.0698$$

The value is positive because $4^\circ$ lands in the first quadrant, where every trigonometric ratio is positive. It is close to zero because $4^\circ$ is only a little way past $0^\circ$, and $\sin 0^\circ = 0$.

Here are the companion ratios at the same angle, each to four decimal places:

• $\sin 4^\circ \approx 0.0698$

• $\cos 4^\circ \approx 0.9976$

• $\tan 4^\circ \approx 0.0699$

Notice that $\sin 4^\circ$ and $\tan 4^\circ$ are almost equal, while $\cos 4^\circ$ is nearly $1$. That pattern is a signature of small angles, and the reason for it appears later in the sine function sections below.

How Do You Find Sin 4 Degrees?

Because $4^\circ$ is not one of the special angles, there is no memorised fraction to recall. There are three honest routes to the value, and they agree.

The first route is the right triangle. Draw a right triangle with one acute angle equal to $4^\circ$. The sine of that angle is the side opposite it divided by the hypotenuse.

$$\sin 4^\circ = \frac{\text{opposite}}{\text{hypotenuse}}$$

For a $4^\circ$ angle the opposite side is tiny next to the hypotenuse, so the ratio is small, and measuring a carefully drawn triangle gives roughly $0.07$. This right-triangle picture is the foundation of every trigonometric ratio, and it connects directly to sin cos tan.

The second route is the unit circle, covered in the next section, where $\sin 4^\circ$ is read straight off the vertical coordinate of a point.

The third route is a calculator or a trigonometric table. Set the calculator to degree mode and enter $\sin 4$, or look up the row for $4^\circ$ in a trigonometric table. Both return $0.0698$. What the calculator is doing behind that single keypress is a power series, shown in the exact-value section below.

Where Does 4 Degrees Sit On The Unit Circle?

On the unit circle, a circle of radius $1$ centred at the origin, an angle is measured anticlockwise from the positive $x$-axis. The point where the angle's ray meets the circle has coordinates $(\cos\theta, \sin\theta)$. So the sine of the angle is simply the height of that point.

For $\theta = 4^\circ = \frac{\pi}{45}$ rad, the point sits at:

$$(\cos 4^\circ, \sin 4^\circ) \approx (0.9976,\ 0.0698)$$

The ray has barely lifted off the $x$-axis, so the point is almost due east, far to the right and only a hair above the horizontal. Its height above the axis, $0.0698$, is $\sin 4^\circ$. Its distance along the axis, $0.9976$, is $\cos 4^\circ$. This is the same value the right triangle gave, now read as a coordinate rather than a ratio.

The unit-circle reading and the right-triangle ratio give the identical number. That double check, one from the triangle and one from the circle, is the surest way to trust a trig value. To explore how the tangent line fits the same picture, see the unit circle with tangent.

Does Sin 4 Degrees Have An Exact Value?

This is where honesty matters. Angles such as $30^\circ$, $45^\circ$, and $60^\circ$ have clean surd values like $\frac{1}{2}$, $\frac{\sqrt{2}}{2}$, and $\frac{\sqrt{3}}{2}$. Sin 4 degrees does not.

An angle of $4^\circ$ is not constructible with the classic compass and straightedge, and $\sin 4^\circ$ cannot be written as a tidy combination of ordinary square roots. Any source that prints a short radical for it is either rounding or inventing. The trustworthy exact statements are these two:

$$\sin 4^\circ = \cos 86^\circ, \qquad \sin 4^\circ = \sin\frac{\pi}{45}$$

The first is the cofunction relationship: the sine of an angle equals the cosine of its complement, and $90^\circ - 4^\circ = 86^\circ$. You can confirm it numerically, since $\cos 86^\circ = 0.0698$ as well. The rule behind it is set out in the cofunction identities and in the trigonometric ratios of complementary angles.

So how does a calculator produce $0.0698$? It uses the Taylor series for sine, which needs the angle in radians. With $x = \frac{\pi}{45}$:

$$\sin x = x - \frac{x^{3}}{6} + \frac{x^{5}}{120} - \cdots$$

The denominators $6$ and $120$ are the factorials of $3$ and $5$. Substituting $x = 0.069813$ term by term gives $0.069813 - 0.0000567 + \dots \approx 0.069756$, which rounds to $0.0698$. A single term is already a good estimate, because for a small angle in radians $\sin x \approx x$, so $\sin 4^\circ \approx \frac{\pi}{45} \approx 0.0698$. Understanding why the input must be in radians starts with what is a radian.

How Does Sin 4 Degrees Compare To Nearby Small Angles?

Small angles line up in an almost straight run, because near $0^\circ$ the sine grows roughly in step with the angle. The table lists each angle in degrees and radians alongside its three main ratios.

Table: Sine, cosine, and tangent of small first-quadrant angles, to four decimal places.

Reading down the sine column, each step of $1^\circ$ adds about $0.0175$, the radian size of one degree. That near-constant step is the small-angle rule in action. For the neighbouring value pages, see sin 1 degrees, sin 2 degrees, and sin 5 degrees. The matching cosine value lives at cos 4 degrees.

Why Is Sin 4 Degrees Positive And Small?

The sign and the size both come from where $4^\circ$ sits on the circle.

• The sign is positive because $4^\circ$ is in the first quadrant. Using the ASTC rule (All ratios positive in quadrant I), sine, cosine, and tangent are all positive there, so $\sin 4^\circ > 0$.

• The value is small because $4^\circ$ is barely above $0^\circ$, and $\sin 0^\circ = 0$. The height of the unit-circle point has only just started to rise.

• The reference angle is $4^\circ$ itself. In the first quadrant the reference angle equals the angle, so no sign flip or subtraction is needed before reading the value.

Put together, $\sin 4^\circ$ has to be a small positive number, and $0.0698$ fits all three observations. For the wider framework of how these ratios are defined, see trigonometric ratios.

Who Discovered How To Compute Sine Values?

Long before calculators, mathematicians built sine values by hand, one angle at a time, into tables that navigators and astronomers trusted for centuries. The tradition runs from ancient Greece through India.

Two earlier figures made the small-angle values of sine computable at all:

• Aryabhata (476–550 CE, India) recorded one of the first sine tables, listing values he called jya, the root of the word "sine" itself, in steps small enough to interpolate angles like $4^\circ$.

• Claudius Ptolemy (around 100–170 CE, Roman Egypt) built a table of chords in the Almagest that is equivalent to a sine table, and used it to model the motion of the planets.

Where Is Sin 4 Degrees Used In The Real World?

A shallow angle like $4^\circ$ shows up wherever a slope, a beam, or a wave stays gentle.

• Ramps and road grades: an accessibility ramp or a highway incline of a few degrees uses the sine to turn a slope angle into the actual height gained over a run.

• Optics and small-angle physics: for tiny angles, engineers rely on $\sin\theta \approx \theta$ in radians, and $4^\circ$ is small enough that this shortcut stays accurate for lenses, mirrors, and pendulums.

• Surveying and construction: a surveyor sighting a distant point a few degrees above the horizon uses the sine to compute the vertical offset from the horizontal distance.

• Signal and sound amplitude: a wave nudged only $4^\circ$ out of phase carries a small sine factor that sets how much two signals reinforce or cancel.

Across all of these, the same small number, $0.0698$, is what converts a shallow angle into a real distance or a real amplitude.

What Are The Most Common Mistakes With Sin 4 Degrees?

These four slips cause most wrong answers involving a small angle like $4^\circ$, and each has a clean fix.

Leaving the calculator in radian mode.

Where it slips in:

A student types $\sin 4$ expecting $0.0698$ but the calculator is set to radians, so it returns $\sin(4\text{ rad}) \approx -0.7568$, a negative number.

Don't do this:

Do not read the display before checking the angle-mode indicator.

The correct way:

Set the mode to degrees for $\sin 4^\circ$, or convert first: $4^\circ = \frac{\pi}{45}$ rad, then enter that radian value in radian mode.

Thinking a small value might be negative.

Where it slips in:

Because $0.0698$ is close to zero, a student second-guesses the sign and writes $-0.0698$.

Don't do this:

Do not confuse "small" with "negative." Size and sign are separate questions.

The correct way:

Check the quadrant. $4^\circ$ is in quadrant I, where sine is positive, so $\sin 4^\circ = +0.0698$.

Mis-pairing the cofunction.

Where it slips in:

A student recalls that sine equals a cosine and writes $\sin 4^\circ = \cos 4^\circ$, which is false since $\cos 4^\circ \approx 0.9976$.

Don't do this:

Do not pair an angle with itself. The cofunction uses the complement.

The correct way:

Subtract from $90^\circ$: $\sin 4^\circ = \cos(90^\circ - 4^\circ) = \cos 86^\circ \approx 0.0698$.

Inventing a surd "exact value."

Where it slips in:

Trained by $30^\circ$ and $45^\circ$, a student writes a made-up radical for $\sin 4^\circ$ to look exact.

Don't do this:

Do not force a clean root onto a non-constructible angle.

The correct way:

State the value as a decimal, $\sin 4^\circ \approx 0.0698$, or as an exact relationship, $\sin 4^\circ = \cos 86^\circ = \sin\frac{\pi}{45}$.

Practice Problems On Sin 4 Degrees

Try each, then check the answer that follows.

1. State $\sin 4^\circ$ to four decimal places.
   (Answer: $0.0698$.)

2. Write $4^\circ$ in radians as a fraction of $\pi$.
   (Answer: $\frac{\pi}{45} \approx 0.0698$ rad.)

3. Use the cofunction identity to rewrite $\sin 4^\circ$ as a cosine.
   (Answer: $\cos 86^\circ$.)

4. Given $\sin 4^\circ \approx 0.0698$ and $\cos 4^\circ \approx 0.9976$, estimate $\tan 4^\circ$.
   (Answer: $\tan 4^\circ = \frac{\sin 4^\circ}{\cos 4^\circ} \approx \frac{0.0698}{0.9976} \approx 0.0699$.)

5. Use the small-angle rule $\sin x \approx x$ to estimate $\sin 4^\circ$ from its radian measure.
   (Answer: $\frac{\pi}{45} \approx 0.0698$, matching the true value to three decimals.)

6. A ramp rises at $4^\circ$ over a run whose hypotenuse is $50$ m. Estimate the height gained.
   (Answer: height $= 50 \times \sin 4^\circ \approx 50 \times 0.0698 = 3.49$ m.)

Where Should You Go Next After Sin 4 Degrees?

Sin 4 degrees is one small window onto the whole sine function, and several doors open from here.

1. The sine function. See how single values like this one join into the full sine curve across every angle.

2. Trigonometric table. Find the ready-reference values for the standard angles alongside small ones like $4^\circ$.

3. Cofunction identities. Master the $\sin\theta = \cos(90^\circ - \theta)$ rule that links $\sin 4^\circ$ to $\cos 86^\circ$.

If your child is building these foundations, a live Bhanzu trainer teaches sine values starting from the unit circle and the right triangle together in the Bhanzu trigonometry program.