What Does Tan 1 Degrees Mean?
Tangent of an angle is the opposite side over 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 $1^\circ$ that point is about $(0.9998, 0.0175)$, so tangent is roughly $\dfrac{0.0175}{0.9998} \approx 0.0175$.
Like $20^\circ$ or $35^\circ$, $1^\circ$ is not a standard angle, so there is no neat fraction or square root equal to $\tan 1^\circ$. Treat it as a calculator or approximation value, not something to memorise. What makes $1^\circ$ special is not an exact form but how well the linear estimate works this close to zero.
Where Does Tan 1 Degrees Show Up?
A $1^\circ$ tilt is a tiny slope, and $\tan 1^\circ \approx 0.0175$ is the gradient it produces: roughly a $1.75%$ grade, close to a gentle wheelchair-ramp minimum. Small deflections like this appear in engineering tolerances, a beam that bends by a degree, or a laser that drifts by a degree over a long distance.
The value also anchors the small-angle approximation that physics leans on for pendulums and optics, where angles near $1^\circ$ let $\tan\theta$, $\sin\theta$, and $\theta$ be treated as the same number. There is even a neat identity: the product $\tan 1^\circ \times \tan 2^\circ \times \cdots \times \tan 89^\circ = 1$, because each $\tan\theta$ pairs with its cofunction $\tan(90^\circ - \theta)$ to give $1$ through the trigonometric ratios.
Standard-Angle Reference Table
Because $1^\circ$ is so close to $0^\circ$, its tangent is tiny, and the small-angle rule $\tan\theta \approx \theta$ (with $\theta$ in radians) is nearly perfect this low. The table shows how the approximation holds at first and starts to drift.
Angle (degrees) | Angle (radians) | $\tan\theta$ | $\theta$ in radians | Rule $\tan\theta \approx \theta$ |
|---|---|---|---|---|
$0^\circ$ | $0$ | $0$ | $0$ | exact |
$1^\circ$ | $\dfrac{\pi}{180}$ | $0.01746$ | $0.01745$ | error $\approx 0.01%$ |
$2^\circ$ | $\dfrac{\pi}{90}$ | $0.03492$ | $0.03491$ | error $\approx 0.04%$ |
$5^\circ$ | $\dfrac{\pi}{36}$ | $0.08749$ | $0.08727$ | error $\approx 0.25%$ |
$10^\circ$ | $\dfrac{\pi}{18}$ | $0.17633$ | $0.17453$ | error $\approx 1%$ |
At $1^\circ$ the radian measure and the tangent agree to four decimal places. By $10^\circ$ they differ by about $1%$, which is the small-angle rule beginning to break down.
How Do You Find The Exact Value Of Tan 1 Degrees?
There is no simple exact value, but at this size the number is easy to pin down.
Method 1: The calculator, in degree mode.
Set the calculator to degrees and enter $\tan(1)$:
$$\tan 1^\circ = 0.01745506\ldots$$
Confirming degree mode matters more here than almost anywhere, because the radian-mode answer is nowhere near.
Method 2: The small-angle rule (near-exact at 1°).
For small angles in radians, $\tan\theta \approx \theta$. Convert first: $1^\circ = \dfrac{\pi}{180} \approx 0.0174533$ radians. Then:
$$\tan 1^\circ \approx \frac{\pi}{180} \approx 0.0174533$$
The true value is $0.0174551$, so the estimate is off by about $0.01%$, effectively exact for most work. The one non-negotiable step is converting to radians; the rule fails if you use the degree number $1$.
Method 3: Sine over cosine.
Using $\sin 1^\circ \approx 0.0174524$ and $\cos 1^\circ \approx 0.9998477$:
$$\tan 1^\circ = \frac{\sin 1^\circ}{\cos 1^\circ} \approx \frac{0.0174524}{0.9998477} = 0.0174551$$
Because $\cos 1^\circ$ is so close to $1$, the tangent is almost identical to the sine at this angle.
Examples Of Tan 1 Degrees
Example 1
A ramp rises at $1^\circ$ over a horizontal run of $40$ m. How much does it climb?
$$\text{rise} = 40 \times \tan 1^\circ \approx 40 \times 0.01746 = 0.698 \text{ m}$$
The climb is about $70$ cm.
Example 2
Estimate $\tan 1^\circ$ using the small-angle rule.
Wrong attempt. A student remembers $\tan\theta \approx \theta$ and plugs in the degree number directly: $\tan 1^\circ \approx 1$.
That is off by a factor of nearly $60$. A calculator gives $0.0175$, not $1$. The value $1$ is actually $\tan 45^\circ$, which should be a clue that plugging the degree count in raw is wrong.
Correct. The rule needs the angle in radians. Convert first:
$$1^\circ = \frac{\pi}{180} \approx 0.01745, \qquad \tan 1^\circ \approx 0.01745$$
That matches the calculator to four decimals. The whole error was skipping the degree-to-radian conversion.
Example 3
Find $2000 \times \tan 1^\circ$, the horizontal drift of a beam that leans $1^\circ$ over a $2000$ mm length.
$$2000 \times \tan 1^\circ \approx 2000 \times 0.017455 = 34.9 \text{ mm}$$
A $1^\circ$ lean over two metres shifts the top by about $35$ mm.
Example 4
Evaluate $\tan 1^\circ \times \tan 89^\circ$.
Since $89^\circ = 90^\circ - 1^\circ$ and $\tan(90^\circ - \theta) = \cot\theta$, the product is:
$$\tan 1^\circ \times \tan 89^\circ = \tan 1^\circ \times \cot 1^\circ = 1$$
The two nearly-reciprocal values multiply to exactly $1$.
Example 5
Confirm the tiny tangent with sine and cosine, using $\sin 1^\circ \approx 0.017452$ and $\cos 1^\circ \approx 0.999848$.
$$\frac{\sin 1^\circ}{\cos 1^\circ} \approx \frac{0.017452}{0.999848} = 0.017455 = \tan 1^\circ$$
The ratio reproduces the value. The same $\sin 1^\circ$ is derived in the article on sin 1 degrees.
Where Students Trip Up On Tan 1 Degrees
Mistake 1: Using degrees in the small-angle rule
Where it slips in: Applying $\tan\theta \approx \theta$ without converting to radians first. Students first meeting the approximation often forget it is a radian rule.
Don't do this: Writing $\tan 1^\circ \approx 1$ by using the degree number as $\theta$.
The correct way: Convert to radians: $1^\circ = \dfrac{\pi}{180} \approx 0.01745$, so $\tan 1^\circ \approx 0.01745$. The rule only works in radians.
Mistake 2: Expecting tan 1° to be near 1
Where it slips in: Reading the "$1$" in $\tan 1^\circ$ as if it should produce a value near $1$, the way $\tan 45^\circ$ does.
Don't do this: Guessing $\tan 1^\circ \approx 1$.
The correct way: $1^\circ$ is a tiny angle just above $0^\circ$, so its tangent is tiny too, about $0.0175$. Values near $1$ belong to angles near $45^\circ$, not $1^\circ$.
Mistake 3: Leaving the calculator in radian mode
Where it slips in: Entering $\tan(1)$ without checking the mode, so the calculator reads $1$ radian instead of $1$ degree.
Don't do this: Trusting a result of about $1.557$, which is $\tan(1\text{ radian})$, for $\tan 1^\circ$.
The correct way: Switch to degree mode, where $\tan(1)$ returns $0.01746$. To enter the degree value in radian mode, use $\tan(\pi/180)$.
Key Takeaways
Tan 1 degree is approximately $0.0175$ (fuller: $0.01745506$), with no simple exact surd form.
$1^\circ$ is not a standard angle, so it is a calculator or approximation value, not one to memorise.
The small-angle rule $\tan\theta \approx \theta$ (in radians) gives $\tan 1^\circ \approx \dfrac{\pi}{180} \approx 0.0175$, accurate to about $0.01%$.
In radians, $\tan 1^\circ = \tan\left(\dfrac{\pi}{180}\right)$, and its most common slip is using the degree number in the small-angle rule.
To build fluency with radians, small angles, and the special values alongside a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or flexible math classes online.
Practice These Before Moving On
A rod leans $1^\circ$ over a $3$ m length. Use $\tan 1^\circ$ to find its horizontal drift in millimetres.
Convert $1^\circ$ to radians and compare $\dfrac{\pi}{180}$ with the true $\tan 1^\circ$; state the percentage error.
Show that $\tan 1^\circ \times \tan 89^\circ = 1$ using a cofunction relationship.
Want a live Bhanzu trainer to walk through more tan 1 degree problems? Book a free demo class.
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