What is the exact value of cos(5π/6) and how do we calculate it?

Decoding the Trigonometric Puzzle: The Exact Value of cos(5\u03c0/6)

In the realm of mathematics, trigonometric values often appear as abstract symbols, yet they represent profound geometric relationships that have been studied for millennia. To find the exact value of $\cos(5\u03c0/6)$, we can break the problem down into fundamental geometric and trigonometric principles.

1. Convert Radians to Degrees (Optional Context)

While mathematicians prefer radians for their natural alignment with calculus, converting to degrees can sometimes build immediate intuition. Since $\u03c0$ radians equal $180^\circ$:

$$\frac{5\u03c0}{6} = \frac{5 \times 180^\circ}{6} = 5 \times 30^\circ = 150^\circ$$

2. Identify the Quadrant and Sign

An angle of $150^\circ$ (or $\frac{5\u03c0}{6}$ radians) lies in the second quadrant of the Cartesian coordinate system (between $90^\circ$ / $\frac{\u03c0}{2}$ and $180^\circ$ / $\u03c0$). In the second quadrant, the x-coordinates are negative. Because the cosine function corresponds to the x-value on the unit circle, cos is negative in the second quadrant.

3. Determine the Reference Angle

The reference angle is the acute angle formed with the x-axis. For $\frac{5\u03c0}{6}$, the reference angle is:

$$\u03c0 - \frac{5\u03c0}{6} = \frac{\u03c0}{6}$$

(In degrees: $180^\circ - 150^\circ = 30^\circ$)

4. Evaluate the Final Value

We know from standard special right triangles that the cosine of $\frac{\u03c0}{6}$ (or $30^\circ$) is $\frac{\sqrt{3}}{2}$.

Applying the negative sign dictated by the second quadrant:

$$\cos\left(\frac{5\u03c0}{6}\right) = -\cos\left(\frac{\u03c0}{6}\right) = -\frac{\sqrt{3}}{2}$$