Sin 1 Degrees : Value 0.0175 and How to Find It

What Does Sin 1 Degrees Mean?

Sine is one of the three trigonometric ratios - in a right triangle it is the side opposite the angle divided by the hypotenuse. For a $1^\circ$ angle the opposite side is very short next to the hypotenuse, so the ratio is small: about $0.0175$.

Because $1^\circ$ is not one of the special angles $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$, its sine has no clean surd form. It is a calculator or approximation value, not something to memorise the way you memorise $\sin 30^\circ = \dfrac{1}{2}$.

Where Does Sin 1 Degrees Show Up?

Tiny angles are the home of $\sin 1^\circ$. In surveying and astronomy, a $1^\circ$ tilt over a long distance produces a small but measurable rise, and $\sin 1^\circ \approx 0.0175$ is the fraction of the distance that becomes height.

The value also anchors the small-angle regime engineers rely on: for angles this small, $\sin\theta$, $\theta$ (in radians), and $\tan\theta$ are almost equal, which simplifies pendulum motion, optics, and beam deflection. That near-equality reads directly off the unit circle, where a $1^\circ$ arc and its vertical height are nearly the same length.

Small-Angle Sine Reference Table

Unlike $30^\circ$ or $45^\circ$, one degree has no tidy exact value - its sine is a decimal you read from a calculator. The table below shows small angles and how closely the radian small-angle rule $\sin\theta \approx \theta$ tracks the true value.

At $1^\circ$ the estimate and the true value match to five decimals. By $10^\circ$ the gap has grown to about half a percent, which is the small-angle rule starting to drift as the angle opens.

How Do You Find The Exact Value Of Sin 1 Degrees?

There is no simple radical to write down, so the honest routes are a good approximation and a calculator check.

Method 1: Radian conversion plus the small-angle rule.

First convert the degree angle to radians, since the small-angle rule works in radians:

$$1^\circ = 1 \times \frac{\pi}{180} \approx 0.0174533 \text{ radians}$$

For angles this small, sine is almost equal to the angle in radians:

$$\sin 1^\circ \approx 0.0174533$$

The true value is $0.0174524$, so this estimate is right to five decimal places. Below about $10^\circ$ to $15^\circ$ this rule is excellent; past that it drifts and should not be trusted.

Method 2: Calculator check (degree mode).

Set the calculator to degree mode and enter $\sin(1)$, which returns:

$$\sin 1^\circ = 0.0174524\ldots$$

A closed radical form does exist - it comes from linking $\sin 1^\circ$ to $\sin 3^\circ$ through a triple-angle relation and solving a cubic - but it is a long nested-root expression with no classroom use. For any real calculation, the decimal above is the value. Compare this with a special angle like sin 15 degrees, which does have a clean surd form.

Examples Of Sin 1 Degrees

Example 1

Estimate $\sin 1^\circ$ using the small-angle rule, then state the true value.

$$1^\circ = \frac{\pi}{180} \approx 0.017453 \text{ rad}, \qquad \sin 1^\circ \approx 0.017453$$

The true value is $0.017452$, so the estimate is accurate to five decimals.

Example 2

A student needs $\sin 1^\circ$ and reasons: "$\sin 30^\circ = \dfrac{1}{2}$, so $\sin 1^\circ$ should be about $\dfrac{1}{2} \times \dfrac{1}{30} = 0.0167$." What is wrong?

Wrong attempt. Scaling $\sin 30^\circ$ down by the ratio of the angles gives $\sin 1^\circ \approx 0.0167$.

That treats sine as directly proportional to the angle, which it is not across a wide range — $\sin 30^\circ = 0.5$, not $30$ times $\sin 1^\circ$. The scaling instinct only works for very small angles, and even then in radians, not degrees.

Correct. Convert to radians and use $\sin\theta \approx \theta$: $\sin 1^\circ \approx 0.017453$, close to the true $0.017452$. The proportionality holds against the radian measure, not the degree number.

Example 3

Find the height gained by a ramp $50$ m long tilted at $1^\circ$.

$$\text{height} = 50 \times \sin 1^\circ \approx 50 \times 0.017452 = 0.873 \text{ m}$$

Example 4

Use the cofunction relationship to write $\sin 1^\circ$ in terms of cosine.

Since $\sin\theta = \cos(90^\circ - \theta)$:

$$\sin 1^\circ = \cos 89^\circ \approx 0.017452$$

The cofunction identity means a tiny sine equals a near-$90^\circ$ cosine.

Example 5

Check the small-angle rule at $10^\circ$ to see it drift.

$$10^\circ = \frac{\pi}{18} \approx 0.174533 \text{ rad}, \qquad \sin 10^\circ = 0.173648$$

The estimate $0.174533$ now differs from the true $0.173648$ by about $0.5$ - visible proof that the rule weakens as the angle grows. See sin 10 degrees for that value in full.

Where Students Trip Up On Sin 1 Degrees

Mistake 1: Applying the small-angle rule in degrees

Where it slips in: Writing $\sin 1^\circ \approx 1$ because "$\sin\theta \approx \theta$."

Don't do this: Using the degree number $1$ as the estimate. The rule $\sin\theta \approx \theta$ only holds when $\theta$ is measured in radians.

The correct way: Convert first: $1^\circ \approx 0.017453$ radians, then $\sin 1^\circ \approx 0.017453$. Skipping the radian conversion is the single most common error with small-angle sines.

Mistake 2: Hunting for a clean radical form

Where it slips in: Assuming every angle has a surd value like $\dfrac{\sqrt{3}}{2}$.

Don't do this: Spending time trying to write $\sin 1^\circ$ as a simple root. Only the special angles have tidy forms.

The correct way: Treat $\sin 1^\circ$ as a calculator value, $0.0174524$. It is an approximation skill, not a memorisation target.

Mistake 3: Trusting the small-angle rule too far

Where it slips in: Using $\sin\theta \approx \theta$ at $30^\circ$ or beyond.

Don't do this: Estimating $\sin 30^\circ \approx 0.524$ radians - the true value is $0.5$, an error of nearly $5$.

The correct way: Keep the rule below about $10^\circ$ to $15^\circ$, where the error stays tiny, and switch to exact values or a calculator for larger angles.

Key Takeaways

• Sin 1 degrees is approximately $0.0174524$ — a non-special angle with no simple radical, so it is a calculator or approximation value.

• The small-angle rule $\sin\theta \approx \theta$ (in radians) gives $0.0174533$, accurate to five decimals, and stays reliable below about $10^\circ$ to $15^\circ$.

• One degree equals $\dfrac{\pi}{180} \approx 0.0174533$ radians, which is why the sine and the radian measure nearly coincide.

• The main slips are applying the rule in degrees and pushing it to large angles where it drifts.

To make sense of approximations with a teacher, explore Bhanzu's trigonometry tutor or high school math tutor sessions, or browse math classes online.

Practice These Before Moving On

1. Estimate $\sin 2^\circ$ with the small-angle rule, then compare with the true $0.034899$.

2. A telescope tilts $1^\circ$; find the vertical shift over a $200$ m sightline using $\sin 1^\circ$.

3. State the largest angle where you would trust $\sin\theta \approx \theta$ and explain why.

Want a live Bhanzu trainer to walk through approximations and radian conversions? Book a free demo class.

Read More

• The trigonometric table of standard angles

• Sin, cos, and tan — the three core ratios

• Converting radians to degrees

• Trigonometric ratios of specific angles

• Value of sin pi across the sine cycle