Sin 20 Degrees : Value ≈ 0.342 and How to Find It

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.

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

1. Use a calculator to find $\sin 20^\circ + \sin 40^\circ$ and compare with $\sin 60^\circ$.

2. A hill climbs at $20^\circ$; over a $300$ m path, find the height gained using $\sin 20^\circ$.

3. 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.