Sin Pi/6 : Exact Value 1/2 on the Unit Circle

What Does Sin Pi/6 Mean?

Sine is one of the three trigonometric ratios: in a right triangle it is the side opposite the angle divided by the hypotenuse. So $\sin\frac{\pi}{6}$ asks what fraction of the hypotenuse the opposite side reaches when the angle is $30^\circ$.

On the unit circle, sine is the $y$-coordinate of the point where the angle's radius meets the circle. Sweeping through $\frac{\pi}{6}$ radians (a radian being the arc-equals-radius angle) lands the radius at $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, so the height, and therefore sin pi/6, is $\frac{1}{2}$.

Where Does Sin Pi/6 Show Up?

A $\frac{\pi}{6}$ tilt (that is, $30^\circ$) is the gentlest of the standard slopes, so sin pi/6 measures the modest vertical share of anything set at $30^\circ$. A ramp built at $30^\circ$ rises half its slope length for every metre travelled, because the vertical rise scales with $\sin 30^\circ = \frac{1}{2}$, which is why $30^\circ$ is a common design angle for gentle inclines.

The value also sets the clean $\frac{1}{2}$ in optics and mechanics whenever a $30^\circ$ angle appears, from a light ray meeting a surface to a force resolved along a shallow incline. Its neat, rational size is exactly why $30^\circ$ turns up so often in textbook setups.

What Is The Standard-Angle Sine Reference Table?

The angle $\frac{\pi}{6}$ radians is the smallest of the common special angles, so its sine is the smallest non-zero value in the table. Reading down in radian order, $\sin\theta$ starts its climb from $0$, and $\frac{\pi}{6}$ takes the first clean step, $\frac{1}{2}$.

Notice the cofunction mirror: sin pi/6 equals the cosine of its complement, since $\sin\frac{\pi}{6} = \cos\frac{\pi}{3}$. The same $\frac{1}{2}$ appears for $30^\circ$ sine and $60^\circ$ cosine.

How Do You Find The Exact Value Of Sin Pi/6?

Two routes reach $\frac{1}{2}$: build it from a triangle, or read it off the circle.

Method 1: The 30-60-90 triangle.

Take an equilateral triangle with side $2$ and drop a perpendicular from one vertex to the opposite side, splitting it into two right triangles with angles $\angle 30^\circ$, $\angle 60^\circ$, and $\angle 90^\circ$.

In one of them the hypotenuse is $2$ and the side opposite the $30^\circ$ angle is $1$ (half of the base that was split).

$$\sin\frac{\pi}{6} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2}$$

Method 2: The unit circle.

Rotate a radius of length $1$ through $\frac{\pi}{6}$ radians.

Its tip lands at $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.

$$\sin\frac{\pi}{6} = y\text{-coordinate} = \frac{1}{2}$$

Method 3: The calculator check.

In radian mode, $\sin(\pi \div 6)$ returns exactly $0.5$. If the screen shows $0.00913$, the calculator is in degree mode and is reading the input as $\frac{\pi}{6} \approx 0.52$ degrees rather than radians.

Examples Of Sin Pi/6

Example 1

Evaluate $4\sin\left(\frac{\pi}{6}\right)$.

$$4\sin\left(\frac{\pi}{6}\right) = 4 \times \frac{1}{2} = 2$$

Example 2

Evaluate $\sin\left(\frac{\pi}{6}\right)$ from the unit-circle point $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.

Wrong attempt. A student reads the point and picks the first coordinate, writing $\sin\frac{\pi}{6} = \frac{\sqrt{3}}{2}$.

That first coordinate is the $x$-value, which is the cosine. Sine is the height, the $y$-coordinate, so grabbing $\frac{\sqrt{3}}{2}$ reports $\cos\frac{\pi}{6}$ by mistake.

Correct. The $y$-coordinate is $\frac{1}{2}$:

$$\sin\frac{\pi}{6} = \frac{1}{2}$$

Example 3

Verify the identity $\sin^2\left(\frac{\pi}{6}\right) + \cos^2\left(\frac{\pi}{6}\right) = 1$.

Since $\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$:

$$\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1$$

The Pythagorean identity holds.

Example 4

A right triangle has a hypotenuse of $12$ cm and a $\frac{\pi}{6}$ angle. Find the side opposite that angle.

$$\sin\frac{\pi}{6} = \frac{\text{opposite}}{12} \implies \text{opposite} = 12 \times \frac{1}{2} = 6 \text{ cm}$$

Example 5

Evaluate $\sin\left(\frac{7\pi}{6}\right)$ using the reference angle.

The angle $\frac{7\pi}{6}$ lies in the third quadrant, where sine is negative, and its reference angle is $\frac{\pi}{6}$.

$$\sin\frac{7\pi}{6} = -\sin\frac{\pi}{6} = -\frac{1}{2}$$

The degree form of this same negative value is worked in sin 210 degrees.

Where Students Trip Up On Sin Pi/6

Mistake 1: Swapping sine and cosine at pi/6

Where it slips in: Reading the unit-circle point $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$ and taking the wrong coordinate.

Don't do this: Writing $\sin\frac{\pi}{6} = \frac{\sqrt{3}}{2}$, which is actually $\cos\frac{\pi}{6}$.

The correct way: Sine is the height (the $y$-coordinate), so sin pi/6 is $\frac{1}{2}$; the $x$-coordinate $\frac{\sqrt{3}}{2}$ is the cosine. Students first meeting the unit circle often read the coordinates left-to-right and grab $x$ first, so anchor on "sine is the vertical one."

Mistake 2: Confusing sin pi/6 with sin pi/3

Where it slips in: Pairing the $\frac{1}{2}$ and the $\frac{\sqrt{3}}{2}$ with the wrong angle.

Don't do this: Writing $\sin\frac{\pi}{6} = \frac{\sqrt{3}}{2}$, which belongs to $\frac{\pi}{3}$.

The correct way: Smaller acute angle, smaller sine. $\frac{\pi}{6}$ is $30^\circ$, so it takes $\frac{1}{2}$; the larger $\frac{\pi}{3}$ takes $\frac{\sqrt{3}}{2}$. Compare the larger partner in sin pi/3.

Mistake 3: Forgetting the quadrant sign for related angles

Where it slips in: Extending sin pi/6 to angles like $\frac{7\pi}{6}$ or $\frac{5\pi}{6}$ without checking the quadrant.

Don't do this: Writing $\sin\frac{7\pi}{6} = \frac{1}{2}$.

The correct way: The reference angle gives the size $\frac{1}{2}$; the quadrant supplies the sign. In the third quadrant sine is negative, so $\sin\frac{7\pi}{6} = -\frac{1}{2}$.

Key Takeaways

• Sin pi/6 equals $\frac{1}{2}$, or $0.5$, an exact value because $\frac{\pi}{6}$ is a standard angle.

• The 30-60-90 triangle gives it as opposite over hypotenuse; the unit circle gives it as the $y$-coordinate at $\frac{\pi}{6}$.

• In degrees, $\sin\frac{\pi}{6} = \sin 30^\circ = \frac{1}{2}$.

• The most common slip is grabbing the $x$-coordinate $\frac{\sqrt{3}}{2}$: that is the cosine, while sine is the height, $\frac{1}{2}$.

To master the standard angles with a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or structured math classes online.

Practice These Before Moving On

1. Evaluate $6\sin\left(\frac{\pi}{6}\right) + \cos\left(\frac{\pi}{3}\right)$.

2. A ramp rises at $\frac{\pi}{6}$ over a slope length of $10$ m. Use $\sin\frac{\pi}{6}$ to find its vertical rise.

3. Evaluate $\sin\left(\frac{5\pi}{6}\right)$ using the reference angle, and state its sign.

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Read More

• Sin 30 degrees — the degree twin of this angle, same value $\frac{1}{2}$.

• The full trigonometric table — every standard angle in one place.

• Cofunction identities — why $\sin\frac{\pi}{6} = \cos\frac{\pi}{3}$.

• Sin, cos, and tan explained — the three ratios at a glance.

• Applications of trigonometry — where these values do real work.