Tan 1 Degrees : Value 0.0175 and How to Find It

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.

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

1. A rod leans $1^\circ$ over a $3$ m length. Use $\tan 1^\circ$ to find its horizontal drift in millimetres.

2. Convert $1^\circ$ to radians and compare $\dfrac{\pi}{180}$ with the true $\tan 1^\circ$; state the percentage error.

3. 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.

Read More

• Tan 20 Degrees - another non-special angle

• Cos 1 Degrees - exact-vs-decimal value

• Tan 30 Degrees - a clean special angle

• Tan 45 Degrees - value and proof