What Is The Value Of Tan 5 Degrees?
Tan 5 degrees is approximately $0.0875$, and carried further it is $0.0874886$. Written with the angle in radians, $\tan 5^\circ = \tan\frac{\pi}{36}$, where $\frac{\pi}{36} \approx 0.0873$. The angle lies in the first quadrant, so the tangent is positive.
Both forms describe the same angle, and by convention every trigonometry article shows them together:
Degrees: $\tan 5^\circ \approx 0.0875$
Radians: $\tan\frac{\pi}{36} \approx 0.0875$, with $\frac{\pi}{36} \approx 0.0873$ rad
Here is the part most pages skip. There is no tidy exact form for $\tan 5^\circ$ built from square roots, the way the trigonometric table gives $\tan 45^\circ = 1$ or $\tan 30^\circ = \frac{1}{\sqrt{3}}$. For $5^\circ$, the exact value is genuinely just the decimal, and the sections below explain why, and where that decimal comes from.
How Do You Find Tan 5 Degrees?
Start the way you would for any angle: locate the quadrant, fix the sign, then read the ratio. For a first-quadrant angle the sign work is short, but the habit is what protects you on harder angles.
The sign comes from the ASTC rule (All, Sine, Tangent, Cosine), which marks where each function is positive as you sweep counter-clockwise from $0^\circ$:
Quadrant I ($0^\circ$ to $90^\circ$): All positive. $5^\circ$ lives here, so $\tan 5^\circ > 0$.
Quadrant II ($90^\circ$ to $180^\circ$): only Sine positive.
Quadrant III ($180^\circ$ to $270^\circ$): only Tangent positive.
Quadrant IV ($270^\circ$ to $360^\circ$): only Cosine positive.
Since $5^\circ$ is already between $0^\circ$ and $90^\circ$, its reference angle is itself, and no sign flip is needed. That leaves the ratio.
By the right-triangle definition, tangent is the opposite side over the adjacent side. Build a right triangle with a $5^\circ$ angle, and $\tan 5^\circ$ is the ratio of the short vertical leg to the long horizontal leg. From the sin cos tan relationship, the same ratio is $\frac{\sin 5^\circ}{\cos 5^\circ}$:
$$\tan 5^\circ = \frac{\sin 5^\circ}{\cos 5^\circ} = \frac{0.0872}{0.9962} \approx 0.0875$$
The numerator is small and the denominator is close to $1$, which is exactly why a five-degree tangent is so close to zero.
Where Does 5 Degrees Sit On The Unit Circle?
On the unit circle, the point at $5^\circ$ has coordinates $(\cos 5^\circ, \sin 5^\circ) = (0.9962, 0.0872)$, and $\tan 5^\circ$ is the $y$-coordinate divided by the $x$-coordinate.
$$\tan 5^\circ = \frac{y}{x} = \frac{0.0872}{0.9962} \approx 0.0875$$
The point sits just above the positive $x$-axis, a whisker off the $3$ o'clock position. Because it has barely lifted off the axis, the height $y$ is tiny while the width $x$ is almost the full radius, and their ratio stays small and positive. This is the same $0.0875$ the triangle gave, which is the whole point of double-anchoring a value: the unit circle with tangent and the right triangle are two views of one number.
Why Does Tan 5 Degrees Have No Simple Exact Value?
The short answer: $5^\circ$ cannot be built from the "constructible" angles with compass-and-straightedge steps, so no finite tower of square roots reproduces it. The value is real and exact, but its exact form is the decimal, not a surd.
Here is the reasoning, kept to the essentials:
Some angles are constructible. Angles like $30^\circ$, $45^\circ$, $60^\circ$, and even $15^\circ$ can be reached by bisecting and combining known angles, so their tangents come out as surds. For instance $\tan 15^\circ = 2 - \sqrt{3}$.
Trisection is the wall. Getting from $15^\circ$ down to $5^\circ$ means cutting an angle into three equal parts, and angle trisection is not possible with the classical tools for a general angle.
The algebra confirms it. The triple-angle relation ties $\tan 5^\circ$ to $\tan 15^\circ$ through a cubic equation, $\tan 15^\circ = \dfrac{3t - t^3}{1 - 3t^2}$ with $t = \tan 5^\circ$. That cubic is irreducible and falls into the casus irreducibilis, the case where the real roots cannot be written with real radicals.
So the decimal is not laziness. It is the honest exact answer for an angle that resists surd form. Compare this with $\tan 45^\circ$, a clean $1$, and you can see the difference between a special angle and an ordinary one at a glance.
Table: Tangent across nearby angles, degrees and radians.
Angle | Radians | $\tan$ (exact) | $\tan$ (4 dp) |
|---|---|---|---|
$0^\circ$ | $0$ | $0$ | $0.0000$ |
$5^\circ$ | $\frac{\pi}{36}$ | no simple surd | $0.0875$ |
$10^\circ$ | $\frac{\pi}{18}$ | no simple surd | $0.1763$ |
$15^\circ$ | $\frac{\pi}{12}$ | $2 - \sqrt{3}$ | $0.2679$ |
$30^\circ$ | $\frac{\pi}{6}$ | $\frac{1}{\sqrt{3}}$ | $0.5774$ |
$45^\circ$ | $\frac{\pi}{4}$ | $1$ | $1.0000$ |
The links go to the value pages that exist: tan 30 degrees, tan 45 degrees, and tan 60 degrees. The neighbours tan 10 degrees and tan 15 degrees sit in the same non-special family as tan 5 degrees.
How Is The Decimal For Tan 5 Degrees Actually Calculated?
If there is no surd, where does $0.0874886$ come from? A calculator does not measure a triangle. It sums a power series, the modern descendant of the hand-built sine tables that trigonometry ran on for two thousand years.
For a small angle $x$ measured in radians, the tangent is well approximated by:
$$\tan x \approx x + \frac{x^3}{3} + \frac{2x^5}{15}$$
Feed in $x = \frac{\pi}{36} \approx 0.0872665$:
$$\tan\frac{\pi}{36} \approx 0.0872665 + \frac{(0.0872665)^3}{3} + \dots \approx 0.0872665 + 0.0002215 \approx 0.0874880$$
Two terms already land within a rounding step of the true $0.0874886$, because each later term is far smaller than the last. That rapid shrinking is why small angles are the easiest to tabulate, and why $\tan 5^\circ$ is barely more than the radian measure $0.0873$ itself. For a tiny angle, the tangent and the angle-in-radians almost coincide.
Who Discovered How To Calculate Values Like Tan 5 Degrees?
Long before calculators, people computed tables of these values by hand, and the story runs across three continents and roughly sixteen centuries.
Two earlier figures built the road Madhava paved:
Hipparchus of Nicaea (c. 190–120 BCE, Greece) is often called the founder of trigonometry, credited with the first known table of chords, the ancestor of the sine table.
Aryabhata (476–550 CE, India) tabulated sine values (which he called jya) at regular intervals, an approach that let astronomers find the trig value of angles between the entries by interpolation.
Where Is Tan 5 Degrees Used In The Real World?
Small tangents describe gentle slopes and slight tilts, and those show up wherever a shallow angle has to be measured or engineered.
Accessibility and roads: ramp gradients and highway grades are shallow angles. A tangent near $0.0875$ describes a rise of about $8.75$ units per $100$ run, close to the steepest a comfortable wheelchair ramp is allowed to be.
Surveying and construction: a theodolite reads small vertical angles to a distant point, and the tangent converts that angle into a height difference over a known distance.
Optics and screens: the slight tilt of a lens, mirror, or display panel is set with small-angle tangents to steer light or reduce glare by a controlled amount.
Astronomy and navigation: the apparent shift of a star or landmark across a small angle is turned into a real distance using its tangent, the backbone of parallax measurement.
One shallow ratio, sized on purpose, quietly decides whether a ramp is legal, a road is safe, and a lens points where it should.
What Are The Most Common Mistakes With Tan 5 Degrees?
These four errors account for most wrong answers on small non-special angles, and each has a clean fix.
Leaving the calculator in radian mode.
Where it slips in:
A student types $5$, presses tan, and reads $-3.38$, then assumes the tangent of five is negative.
Don't do this:
Do not trust the number before checking the angle mode. In radian mode the calculator computed $\tan(5\text{ rad})$, a completely different angle in the third quadrant, not $\tan 5^\circ$.
The correct way:
Set the mode to degrees for $\tan 5^\circ$, or convert first: $5^\circ = \frac{\pi}{36}$ rad, then take the tangent. Either route gives $0.0875$.
Expecting a special-angle shortcut.
Where it slips in:
A student hunts for a surd form for $\tan 5^\circ$, or rounds $5^\circ$ up to $0^\circ$ or down to nothing, expecting a neat answer.
Don't do this:
Do not force a clean radical. $5^\circ$ is not constructible, so no simple surd exists, and rounding the angle changes the value.
The correct way:
Use the decimal $0.0875$ (or more places if needed), and reach for a series or a table, exactly as the calculation section shows.
Getting the quadrant sign wrong.
Where it slips in:
A student carries a sign habit from a second- or third-quadrant problem and writes $\tan 5^\circ$ as negative.
Don't do this:
Do not attach a sign without placing the angle. $5^\circ$ is in Quadrant I, where every ratio is positive.
The correct way:
Run ASTC first. All functions are positive in Quadrant I, so $\tan 5^\circ = +0.0875$.
Confusing the cofunction partner.
Where it slips in:
A student recalls a cofunction rule and writes $\tan 5^\circ = \cot 5^\circ$, pairing the angle with itself.
Don't do this:
Do not pair an angle with its own cotangent. Cofunctions pair an angle with its complement, the angle that completes $90^\circ$.
The correct way:
The complement of $5^\circ$ is $85^\circ$, so $\tan 5^\circ = \cot 85^\circ$. The cofunction identities and the trigonometric ratios of complementary angles spell this out in full.
Practice Problems On Tan 5 Degrees
Work each one, then check against the answer.
State $\tan 5^\circ$ to four decimal places.
(Answer: $0.0875$.)Write the angle $5^\circ$ in radians as a fraction of $\pi$.
(Answer: $\frac{\pi}{36} \approx 0.0873$ rad.)Use the cofunction relation to rewrite $\tan 5^\circ$ with cotangent.
(Answer: $\tan 5^\circ = \cot 85^\circ$.)Given $\sin 5^\circ = 0.0872$ and $\cos 5^\circ = 0.9962$, compute $\tan 5^\circ$.
(Answer: $\frac{0.0872}{0.9962} \approx 0.0875$.)Is $\tan 5^\circ$ positive or negative, and why?
(Answer: positive, because $5^\circ$ is in Quadrant I where all ratios are positive.)Using $\tan x \approx x + \frac{x^3}{3}$ with $x = 0.0873$, estimate $\tan 5^\circ$.
(Answer: $0.0873 + 0.00022 \approx 0.0875$.)
Where Should You Go Next After Tan 5 Degrees?
Tan 5 degrees is a doorway into how trigonometry handles ordinary, non-special angles, and a few natural next steps open from here.
Tangent function. See how the tangent behaves across every angle, its graph, its period, and where it shoots to infinity.
Trigonometric table. Compare $\tan 5^\circ$ with the clean special-angle values in one reference, and see the pattern of which angles get surds.
Cofunction identities. Understand the $\tan 5^\circ = \cot 85^\circ$ link and the family of complement rules it belongs to.
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 every number) in the Bhanzu trigonometry program.
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