What Does Tan 20 Degrees Mean?
Tangent of an angle is the ratio of the opposite side to the adjacent side in a right triangle, or on the unit circle, the $y$-coordinate divided by the $x$-coordinate of the point where the angle's radius meets the circle. At $20^\circ$ that point is about $(0.940, 0.342)$, so tangent is roughly $\dfrac{0.342}{0.940} \approx 0.364$.
The honest part: $20^\circ$ is not one of the standard angles, so there is no neat fraction or square root that equals $\tan 20^\circ$ exactly. That is not a gap in your knowledge; it is a property of the number. Treat $\tan 20^\circ$ as a calculator or approximation skill, not something to memorise the way you memorise $\tan 30^\circ = \dfrac{1}{\sqrt{3}}$.
Where Does Tan 20 Degrees Show Up?
Tangent measures slope, so $\tan 20^\circ \approx 0.364$ is the gradient of anything pitched at $20^\circ$: a moderately steep roof, a wheelchair ramp near its practical limit, a hillside road. A $20^\circ$ incline rises about $0.36$ units for every $1$ unit of horizontal run.
The value turns up wherever a real, non-textbook angle is measured: surveying a slope, aiming a solar panel, setting a camera tilt. Because $20^\circ$ is not a special angle, these applications reach for a calculator or the inverse tangent rather than a memorised surd, using the same trigonometric ratios that define every angle.
Standard-Angle Reference Table
Placing $20^\circ$ next to the standard angles shows exactly where it falls: between the exact values of $0^\circ$ and $30^\circ$, but without a clean form of its own.
Angle (degrees) | Angle (radians) | $\tan\theta$ | Exact? |
|---|---|---|---|
$0^\circ$ | $0$ | $0$ | exact |
$10^\circ$ | $\dfrac{\pi}{18}$ | $0.1763$ | no simple form |
$20^\circ$ | $\dfrac{\pi}{9}$ | $0.3640$ | no simple form |
$30^\circ$ | $\dfrac{\pi}{6}$ | $\dfrac{1}{\sqrt{3}} \approx 0.5774$ | exact |
$45^\circ$ | $\dfrac{\pi}{4}$ | $1$ | exact |
The special angles $0^\circ$, $30^\circ$, and $45^\circ$ have exact values worth memorising; $10^\circ$ and $20^\circ$ do not. For the full set of clean values, see the trigonometric table.
How Do You Find The Exact Value Of Tan 20 Degrees?
The short answer is that there is no simple exact value, but there are reliable ways to get the number and understand it.
Method 1: The calculator, in degree mode.
Set the calculator to degrees and enter $\tan(20)$:
$$\tan 20^\circ = 0.36397023\ldots$$
Confirming degree mode is the whole game here; radian mode gives a different number entirely.
Method 2: Why no clean surd exists.
$20^\circ$ is one-third of $60^\circ$, and $\tan 60^\circ = \sqrt{3}$. The triple-angle relationship links them:
$$\tan 60^\circ = \frac{3\tan 20^\circ - \tan^3 20^\circ}{1 - 3\tan^2 20^\circ} = \sqrt{3}$$
Solving that for $\tan 20^\circ$ gives a cubic equation with no solution expressible in simple square roots. This is why $\tan 20^\circ$ stays a decimal while tan 60 degrees has the tidy value $\sqrt{3}$.
Method 3: The small-angle estimate (and its limit).
For small angles measured in radians, $\tan\theta \approx \theta$. Since $20^\circ = \dfrac{\pi}{9} \approx 0.349$ radians:
$$\tan 20^\circ \approx 0.349$$
The true value is $0.364$, so the estimate is low by about $4%$. That is the warning built into this method: the approximation is good below roughly $10^\circ$ and drifts as the angle grows, so by $20^\circ$ it is only a rough check, not the answer.
Examples Of Tan 20 Degrees
Example 1
A ramp rises at $20^\circ$ over a horizontal run of $5$ m. How much does it climb?
$$\text{rise} = 5 \times \tan 20^\circ \approx 5 \times 0.3640 = 1.82 \text{ m}$$
Example 2
Find $\tan 20^\circ$ from the known value $\tan 60^\circ = \sqrt{3}$.
Wrong attempt. A student reasons that since $20^\circ$ is $60^\circ$ divided by $3$, the tangent must divide by $3$ too, and writes $\tan 20^\circ = \dfrac{\sqrt{3}}{3} \approx 0.577$.
Check that against a calculator, which gives $0.364$, not $0.577$. Tangent is not linear in the angle, so dividing the angle by $3$ does not divide the tangent by $3$.
Correct. The real link is the triple-angle formula, not simple division:
$$\tan 60^\circ = \frac{3\tan 20^\circ - \tan^3 20^\circ}{1 - 3\tan^2 20^\circ}$$
Solving numerically returns $\tan 20^\circ \approx 0.364$. The value $0.577$ is actually $\tan 30^\circ$, which is why the shortcut felt plausible.
Example 3
Estimate the height of a tree if its top sits at a $20^\circ$ angle of elevation from a point $30$ m away on level ground.
$$\text{height} = 30 \times \tan 20^\circ \approx 30 \times 0.3640 = 10.9 \text{ m}$$
The tree is about $10.9$ m tall.
Example 4
Evaluate $10\tan 20^\circ$ to three decimal places.
$$10\tan 20^\circ \approx 10 \times 0.36397 = 3.640$$
Example 5
Confirm $\tan 20^\circ = \dfrac{\sin 20^\circ}{\cos 20^\circ}$ using $\sin 20^\circ \approx 0.3420$ and $\cos 20^\circ \approx 0.9397$.
$$\frac{\sin 20^\circ}{\cos 20^\circ} \approx \frac{0.3420}{0.9397} = 0.3640 = \tan 20^\circ$$
The ratio of the two decimals reproduces the tangent, as it must. The same $\sin 20^\circ$ appears in the article on sin 20 degrees.
Where Students Trip Up On Tan 20 Degrees
Mistake 1: Assuming the tangent scales with the angle
Where it slips in: Trying to build $\tan 20^\circ$ from a known angle by dividing. Students first meeting non-special angles often expect tangent to behave like simple proportion.
Don't do this: Writing $\tan 20^\circ = \dfrac{1}{3}\tan 60^\circ$ or $\tan 20^\circ = \dfrac{1}{2}\tan 40^\circ$.
The correct way: Tangent is not proportional to the angle. For a non-special angle, use a calculator or the triple-angle relationship, not scaling.
Mistake 2: Leaving the calculator in radian mode
Where it slips in: Entering $\tan(20)$ without checking the mode. In radian mode the calculator reads $20$ radians, not $20$ degrees.
Don't do this: Trusting a result of about $2.237$, which is $\tan(20\text{ radians})$, for $\tan 20^\circ$.
The correct way: Switch to degree mode first. Then $\tan(20)$ returns $0.3640$, and $\tan 20^\circ$ in radians would be entered as $\tan(\pi/9)$.
Mistake 3: Hunting for an exact surd that isn't there
Where it slips in: Assuming every angle on a homework sheet must have a clean radical form like the special angles.
Don't do this: Forcing $\tan 20^\circ$ into a made-up surd such as $\dfrac{\sqrt{2}}{4}$ to look exact.
The correct way: Accept the decimal. $20^\circ$ is not a standard angle, so $0.3640$ (or the fuller $0.36397023$) is the value. Rounding is expected, not an error.
Key Takeaways
Tan 20 degrees is approximately $0.3640$ (fuller: $0.36397023$), and it has no simple exact surd form.
$20^\circ$ is not a standard angle, so it is a calculator or approximation value, not one to memorise.
In radians, $\tan 20^\circ = \tan\left(\dfrac{\pi}{9}\right)$, and the small-angle estimate $\tan\theta \approx \theta$ gives only $0.349$, low by about $4%$.
Tangent is not proportional to the angle, so $\tan 20^\circ$ cannot be built by dividing $\tan 60^\circ$.
To get comfortable with special and non-special angles alongside a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or flexible math classes online.
Practice These Before Moving On
A slope is inclined at $20^\circ$. Find its rise over a $15$ m run using $\tan 20^\circ$.
Compare the small-angle estimate $\dfrac{\pi}{9}$ with the true $\tan 20^\circ$ and state the percentage error.
Explain in one sentence why $\tan 20^\circ$ has no clean surd form while $\tan 30^\circ$ does.
Want a live Bhanzu trainer to walk through more tan 20 degrees problems? Book a free demo class.
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