Tan 4 Degrees: Value, Radians & How To Find It

What Is The Value Of Tan 4 Degrees?

The value of tan 4 degrees is approximately $0.0699$, and to more decimal places $\tan 4^\circ \approx 0.0699268$. The same angle written in radians is $4^\circ = \frac{\pi}{45} \approx 0.0698$, so you may also see it as $\tan\frac{\pi}{45}$.

The value is positive and small. It is positive because $4^\circ$ lands in the first quadrant, where every trigonometric ratio is positive. It is small because $4^\circ$ is a shallow angle, close to $0^\circ$, and $\tan 0^\circ = 0$.

$$\tan 4^\circ \approx 0.0699 \qquad \left(4^\circ = \tfrac{\pi}{45} \approx 0.0698 \text{ radians}\right)$$

Unlike neat special angles such as tan 30 degrees or tan 45 degrees, the value of $\tan 4^\circ$ is not a tidy fraction or surd. The rest of this page shows where the number comes from and why there is no cleaner form.

How Do You Find Tan 4 Degrees?

The tangent of any angle is the sine of that angle divided by its cosine, and $4^\circ$ is no exception. This is the definition to lean on when an angle has no special-triangle shortcut.

$$\tan 4^\circ = \frac{\sin 4^\circ}{\cos 4^\circ} = \frac{0.069756}{0.997564} \approx 0.069927$$

Three checks confirm the answer is set up correctly:

• Quadrant and sign (ASTC). $4^\circ$ lies in the first quadrant, so by the ASTC rule (All ratios positive in quadrant I) the answer must be positive. A negative result would signal a sign or quadrant error.

• Reference angle. For an angle already between $0^\circ$ and $90^\circ$, the reference angle is the angle itself, so the reference angle of $4^\circ$ is $4^\circ$. There is no reflection or shift to undo.

• Size sanity check. Because $4^\circ$ is close to $0^\circ$, the answer should be close to $\tan 0^\circ = 0$. A value near $0.07$ passes; a value near $1$ would mean the calculator was in the wrong mode.

You can build both inputs from a right triangle or read them off the unit circle. Because tangent is a trigonometric ratio, the right-triangle picture is the first anchor: in a right triangle with a $4^\circ$ angle, $\tan 4^\circ$ is the side opposite the angle divided by the side adjacent to it.

Where Does 4 Degrees Sit On The Unit Circle?

On the unit circle, an angle is measured anticlockwise from the positive $x$-axis, and the point where the angle's ray meets the circle has coordinates $(\cos\theta, \sin\theta)$. For $4^\circ$ that point is almost due east, only just above the axis.

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

The tangent is the $y$-coordinate divided by the $x$-coordinate of that point:

$$\tan 4^\circ = \frac{y}{x} = \frac{\sin 4^\circ}{\cos 4^\circ} \approx \frac{0.0698}{0.9976} \approx 0.0699$$

This matches the right-triangle answer exactly, which is the point of learning both pictures: the ratio is the same number whether you read it from a triangle's sides or from a point on the circle. For a fuller view of how the tangent behaves all the way around, see the unit circle with tangent.

Is There An Exact Value For Tan 4 Degrees?

There is no simple exact value for $\tan 4^\circ$. The angles that have clean closed forms, built from square roots, are the ones you can construct with a compass and straightedge, and those turn out to be the multiples of $3^\circ$. Since $4^\circ$ is not a multiple of $3^\circ$, it falls outside that family, and no finite expression in ordinary square roots produces it.

Being honest about this matters. The number $0.0699268$ is exact as far as it goes, but it is a decimal approximation, not a surd like $\frac{1}{\sqrt{3}}$. Three legitimate ways produce it:

• Cofunction relation. By the cofunction identity, $\tan\theta = \cot(90^\circ - \theta)$, so $\tan 4^\circ = \cot 86^\circ$. This does not simplify the number, but it links $4^\circ$ to its complement $86^\circ$. See the cofunction identities and the trigonometric ratios of complementary angles for the full set.

• Small-angle approximation. For a small angle measured in radians, $\tan x \approx x$. Converting first, $4^\circ = \frac{\pi}{45}$, gives a quick estimate.

• Power series. The tangent function has an infinite series that a calculator uses to reach any precision you need.

The small-angle estimate takes one line:

$$\tan 4^\circ \approx \frac{\pi}{45} \approx 0.06981$$

The power series then sharpens it. The series for tangent, with $x$ in radians, begins:

$$\tan x = x + \frac{x^{3}}{3} + \frac{2x^{5}}{15} + \frac{17x^{7}}{315} + \cdots$$

Keeping the first two terms with $x = \frac{\pi}{45}$:

$$\tan\frac{\pi}{45} \approx \frac{\pi}{45} + \frac{1}{3}\left(\frac{\pi}{45}\right)^{3} \approx 0.06981 + 0.00011 = 0.06993$$

That already rounds to $0.0699$, matching the sine-over-cosine result. Adding more terms drives it toward $0.0699268$. This is why tables and calculators, not clever algebra, are the practical route to $\tan 4^\circ$.

Table: Tangent values for a range of first-quadrant angles, degrees and radians.

Reading down the table, the tangent climbs slowly at first and then faster, which is exactly why a $4^\circ$ tangent is so much smaller than a $30^\circ$ one. A full listing lives in the trigonometric table.

Why Is Tan 4 Degrees Positive And So Small?

Two features of $\tan 4^\circ$ deserve a reason, not just a number: its sign and its size.

• Why positive. In the first quadrant both $\sin 4^\circ$ and $\cos 4^\circ$ are positive, and a positive divided by a positive is positive. Every ratio, sine, cosine, and tangent, is positive for angles between $0^\circ$ and $90^\circ$.

• Why small. Tangent measures how fast height grows relative to width as the angle opens. At $4^\circ$ the ray on the unit circle has barely lifted off the $x$-axis, so the height ($0.0698$) is tiny while the width ($0.9976$) is almost the full radius. A tiny height over a near-full width is a small ratio.

• Why close to the radian measure. For small angles the arc, the height, and the tangent are nearly equal, so $\tan 4^\circ \approx \frac{\pi}{45} \approx 0.0698$. The gap between $0.0698$ and $0.0699$ is the small curvature correction the series adds back.

The tangent function grows without bound as the angle approaches $90^\circ$, but near $0^\circ$ it behaves almost like a straight line through the origin. A $4^\circ$ angle sits deep in that gentle, near-linear zone.

Who Discovered How To Compute Tan 4 Degrees?

No single person found $\tan 4^\circ$. The value is the end of a long chain of table-makers who worked out how to fill the gaps between the constructible angles, and the sharpest tool in that chain is the power series.

Two earlier figures built the tables Madhava's series would eventually surpass:

• Aryabhata (c. 476–550 CE, India) compiled one of the first sine tables, the jya table, in his Aryabhatiya of 499 CE, listing values in steps of $3.75^\circ$.

• Hipparchus (c. 190–120 BCE, Greece) is credited with the first trigonometric table of all, a table of chords, the tool that Ptolemy later refined in the Almagest to interpolate values for fine angles.

Where Is Tan 4 Degrees Used In The Real World?

A small tangent like $\tan 4^\circ$ shows up wherever a gentle slope or a slight tilt has to be measured precisely.

• Accessibility ramps. Building codes cap ramp gradients near a $1$-in-$12$ slope, which is about $4.8^\circ$, so tangents of small angles set how much rise is allowed over a given run.

• Road and rail gradients. Engineers describe a shallow grade as a percentage, and that percentage is the tangent of the slope angle times $100$, so a $4^\circ$ climb is roughly a $7$ grade.

• Camera and antenna tilt. Aiming a security camera or a satellite dish a few degrees off horizontal uses small-angle tangents to convert the tilt into a vertical offset at a distance.

• Surveying and levelling. Reading a small elevation change across a long baseline is a small-angle tangent calculation, the same $\frac{\text{rise}}{\text{run}}$ that defines $\tan 4^\circ$.

The thread through all four is the same ratio: a small angle turns into a small, predictable rise over a known distance.

What Are The Most Common Mistakes With Tan 4 Degrees?

These three errors account for most wrong answers with small-angle tangents, and each has a clean fix.

Leaving the calculator in the wrong angle mode.

Where it slips in:

A student types tan(4) expecting $\tan 4^\circ$, but the calculator is set to radians and returns $\tan 4 \text{ rad} \approx 1.1578$.

Don't do this:

Do not assume the default mode is degrees. The two answers, $0.0699$ and $1.1578$, are nowhere near each other.

The correct way:

Set the mode to degrees before typing, or convert first and compute $\tan\frac{\pi}{45}$ in radian mode. A first-quadrant tangent this small must be near $0.07$, so a result above $1$ is a mode error.

Turning the cofunction into the wrong angle.

Where it slips in:

A student recalls that tangent has a cofunction partner and writes $\tan 4^\circ = \tan 86^\circ$ instead of $\tan 4^\circ = \cot 86^\circ$.

Don't do this:

Do not keep the same function name when switching to the complement. $\tan 86^\circ \approx 14.3$, which is the reciprocal of $\tan 4^\circ$, not equal to it.

The correct way:

Use $\tan\theta = \cot(90^\circ - \theta)$, so $\tan 4^\circ = \cot 86^\circ$. The function name flips from tangent to cotangent when the angle flips to its complement.

Expecting an exact surd form.

Where it slips in:

A student searches for a fraction or root equal to $\tan 4^\circ$, the way $\tan 30^\circ = \frac{1}{\sqrt{3}}$, and treats the decimal as a failure to simplify.

Don't do this:

Do not force a closed form that does not exist. $4^\circ$ is not a multiple of $3^\circ$, so no simple surd represents it.

The correct way:

Report the decimal, $\tan 4^\circ \approx 0.0699$, and state that the value comes from a table, calculator, or series. That is the honest and complete answer.

Practice Problems On Tan 4 Degrees

Work each one, then check against the answer that follows.

1. Convert $4^\circ$ to radians.
   (Answer: $4^\circ = \frac{\pi}{45} \approx 0.0698$ radians.)

2. Write $\tan 4^\circ$ as a cotangent using the cofunction identity.
   (Answer: $\tan 4^\circ = \cot 86^\circ$.)

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

4. Is $\tan 4^\circ$ positive or negative, and in which quadrant does $4^\circ$ lie?
   (Answer: positive, first quadrant.)

5. Use the small-angle approximation to estimate $\tan 4^\circ$.
   (Answer: $\tan 4^\circ \approx \frac{\pi}{45} \approx 0.0698$.)

6. A ramp rises at $4^\circ$ over a horizontal run of $5$ m. Find the vertical rise.
   (Answer: rise $= 5\tan 4^\circ \approx 5 \times 0.0699 \approx 0.35$ m.)

Where Should You Go Next After Tan 4 Degrees?

A small non-special angle like $4^\circ$ is a good doorway into the machinery behind every trigonometric value.

1. Tangent function. See how tangent behaves across the whole circle, from near-zero at $4^\circ$ to undefined at $90^\circ$.

2. Sin cos tan. Anchor tangent back to sine and cosine, the ratio that produced $\tan 4^\circ$ in the first place.

3. Cofunction identities. Understand why $\tan 4^\circ = \cot 86^\circ$ and how every ratio pairs with its complement.

If your child is building these foundations, a live Bhanzu trainer teaches trigonometric values starting from the "why", the unit circle and the right triangle behind each number, in the Bhanzu trigonometry program.