What Does Sin 20 Degrees Mean?
Sine is one of the three trigonometric ratios: the side opposite the angle over the hypotenuse in a right triangle. For a $20^\circ$ angle the opposite side is a little over a third of the hypotenuse, giving about $0.342$.
On the unit circle (radius $1$, centred at the origin), the sine of an angle is the y-coordinate of the point the radius reaches. Rotating $20^\circ$ above the positive x-axis lands the radius near $(0.940, 0.342)$. The y-coordinate, and therefore $\sin 20^\circ$, is about $0.342$, the same point that fixes sin, cos, and tan at $20^\circ$.
Where Does Sin 20 Degrees Show Up?
A roof pitched at $20^\circ$, a wheelchair ramp near its gentlest usable grade, or a projectile launched at $20^\circ$ all use $\sin 20^\circ$ to convert an angle into a vertical rise. A regular eighteen-sided polygon has a central angle of $20^\circ$, so the value shows up in that construction too. Whenever the applications of trigonometry turn a measured incline into a height, an angle like $20^\circ$ is exactly the kind that needs a decimal rather than a neat surd.
Standard-Angle Reference Table
Twenty degrees is not one of the special angles ($0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$) whose sine has a memorable surd. It sits between $15^\circ$ and $30^\circ$, so its sine is a decimal between $0.259$ and $0.5$, read from a calculator. Here are nearby first-quadrant angles as decimals, with their radian measure.
Angle (degrees) | Angle (radians) | $\sin\theta$ (decimal) |
|---|---|---|
$0^\circ$ | $0.0000$ | $0.0000$ |
$10^\circ$ | $0.1745$ | $0.1736$ |
$15^\circ$ | $0.2618$ | $0.2588$ |
$20^\circ$ | $0.3491$ | $0.3420$ |
$30^\circ$ | $0.5236$ | $0.5000$ |
$45^\circ$ | $0.7854$ | $0.7071$ |
At $2^\circ$ the radian measure and the sine agree to four decimals, but by $20^\circ$ the gap has opened: the radian value $0.3491$ overshoots the true sine $0.3420$ by about $2%$. So the small-angle shortcut is already fraying here, and a calculator or the trigonometric table is the reliable source. The complementary value appears on the cos 20 degrees page, where cosine is close to $0.940$.
How Do You Find The Exact Value Of Sin 20 Degrees?
Because $20^\circ$ is not a special angle, no simple surd captures it. Three routes give the value.
Method 1: The calculator.
Set the calculator to degree mode and enter $\sin(20)$:
$$\sin 20^\circ = 0.34202014\ldots$$
Method 2: The triple-angle equation.
The identity $\sin 3\theta = 3\sin\theta - 4\sin^3\theta$ links $20^\circ$ to the special angle $60^\circ$. Setting $\theta = 20^\circ$ makes $3\theta = 60^\circ$:
$$\sin 60^\circ = 3\sin 20^\circ - 4\sin^3 20^\circ$$
Writing $x = \sin 20^\circ$ and using $\sin 60^\circ = \frac{\sqrt{3}}{2}$:
$$4x^3 - 3x + \frac{\sqrt{3}}{2} = 0$$
This cubic has no solution expressible in real square-root radicals (it is the classic "irreducible" case), so $\sin 20^\circ$ can only be pinned down numerically: $x \approx 0.342$. The point of the derivation is not a tidy formula but the honest reason one does not exist.
Method 3: The cofunction relationship.
Sine and cosine are cofunctions, so $\sin 20^\circ = \cos(90^\circ - 20^\circ) = \cos 70^\circ$. Both equal about $0.342$, so a cosine table gives the same value.
Examples Of Sin 20 Degrees
Example 1
Evaluate $50\sin 20^\circ$.
$$50\sin 20^\circ = 50 \times 0.342 = 17.1$$
Example 2
Estimate $\sin 20^\circ$ with the small-angle rule, then judge the estimate.
Wrong attempt. A student converts $20^\circ = \frac{\pi}{9} \approx 0.349$ radians and reports $\sin 20^\circ \approx 0.349$, treating the small-angle rule as exact.
Check it against a calculator: the true value is $0.342$, so the estimate is high by $0.007$, about a $2%$ error. At $2^\circ$ that shortcut was excellent, but $20^\circ$ is already too big for it.
Correct. The dependable value is $\sin 20^\circ \approx 0.342$ from a calculator; the radian $0.349$ is only a rough estimate here. Trusting $\sin\theta \approx \theta$ past about $15^\circ$ is where the error creeps in.
Example 3
A ramp rises at $20^\circ$ over a slope length of $12$ m. Its vertical rise is $12\sin 20^\circ$. Find it.
$$\text{rise} = 12 \times \sin 20^\circ \approx 12 \times 0.342 = 4.104 \text{ m}$$
Example 4
Show that $\sin 20^\circ = \cos 70^\circ$.
By the cofunction identity, $\sin\theta = \cos(90^\circ - \theta)$, so $\sin 20^\circ = \cos(90^\circ - 20^\circ) = \cos 70^\circ$. Both are about $0.342$.
Example 5
Verify that $\sin 20^\circ$ satisfies $4x^3 - 3x + \frac{\sqrt{3}}{2} = 0$ numerically.
Substitute $x = 0.342$:
$$4(0.342)^3 - 3(0.342) + 0.8660 = 0.1600 - 1.0260 + 0.8660 \approx 0.0000$$
The near-zero result confirms $\sin 20^\circ \approx 0.342$ solves the triple-angle cubic.
Where Students Trip Up On Sin 20 Degrees
Mistake 1: Trusting the small-angle rule at 20°
Where it slips in: Reusing $\sin\theta \approx \theta$ (radians) that worked beautifully for tiny angles.
Don't do this: Reporting $\sin 20^\circ \approx 0.349$ as if it were exact.
The correct way: By $20^\circ$ the rule overshoots by about $2%$; the true value is $0.342$. The habit that fixes this is limiting $\sin\theta \approx \theta$ to angles under roughly $15^\circ$.
Mistake 2: Expecting a clean fraction or surd
Where it slips in: Treating $\sin 20^\circ$ like $\sin 30^\circ$ and searching for an exact form.
Don't do this: Writing $\sin 20^\circ = \frac{1}{3}$ or an invented radical.
The correct way: $20^\circ$ is non-special; its sine has no simple surd, so the decimal $0.342$ is the honest value. Check whether the angle is standard before hunting for an exact form.
Mistake 3: Leaving the calculator in radian mode
Where it slips in: Entering $\sin(20)$ with the calculator set to radians.
Don't do this: Reading $\sin(20) \approx 0.913$ and calling it $\sin 20^\circ$.
The correct way: $0.913$ is the sine of $20$ radians, a different angle. Switch to degree mode; $\sin 20^\circ \approx 0.342$.
Key Takeaways
Sin 20 degrees is approximately $0.342$; as a non-special angle it has no clean fraction or surd.
The triple-angle equation $4x^3 - 3x + \frac{\sqrt{3}}{2} = 0$ ties it to $\sin 60^\circ$ but yields no simple radical, so a calculator is needed.
In radians the angle is $\frac{\pi}{9}$, and by cofunctions $\sin 20^\circ = \cos 70^\circ$.
The common slips are over-trusting the small-angle rule at $20^\circ$, expecting a surd, and radian-mode errors.
To go further with a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or math classes online.
Practice These Before Moving On
Use a calculator to find $\sin 20^\circ + \sin 40^\circ$ and compare with $\sin 60^\circ$.
A hill climbs at $20^\circ$; over a $300$ m path, find the height gained using $\sin 20^\circ$.
Explain in one line why the small-angle rule works for $\sin 2^\circ$ but not for $\sin 20^\circ$.
Want a live Bhanzu trainer to walk through more sin 20 degrees problems? Book a free demo class.
Read More
Trigonometric functions — how sine behaves as a function across every angle.
Sin 2 degrees — a tiny angle where the small-angle rule is near-exact.
Sin 15 degrees — a nearby angle that does have an exact surd form.
Cofunction identities — why $\sin 20^\circ = \cos 70^\circ$.
Sum and difference formulas — build non-special values from known angles.
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