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Trigonometry Angles — Pi/6 (30°) — Definition, Formula & Examples

Pi/6 (30°) is one of the standard angles in trigonometry whose sine, cosine, and tangent values can be expressed as exact fractions involving square roots. At this angle, sin⁡=12\sin = \tfrac{1}{2}, cos⁡=32\cos = \tfrac{\sqrt{3}}{2}, and tan⁡=33\tan = \tfrac{\sqrt{3}}{3}.

The angle π6\frac{\pi}{6} radians corresponds to 30° in degree measure. In a 30-60-90 right triangle with hypotenuse of length 1, the side opposite the 30° angle has length 12\frac{1}{2} and the side adjacent has length 32\frac{\sqrt{3}}{2}, yielding the exact trigonometric ratios for all six functions.

Key Formula

\sin\frac{\pi}{6} = \frac{1}{2}, \quad \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}, \quad \tan\frac{\pi}{6} = \frac{\sqrt{3}}{3}$$ $$\csc\frac{\pi}{6} = 2, \quad \sec\frac{\pi}{6} = \frac{2\sqrt{3}}{3}, \quad \cot\frac{\pi}{6} = \sqrt{3}
Where:
  • π6\frac{\pi}{6} = The angle, equal to 30°

How It Works

Place a 30° angle in standard position on the unit circle. The terminal side intersects the circle at the point (32, 12)\left(\frac{\sqrt{3}}{2},\, \frac{1}{2}\right). The xx-coordinate gives cosine and the yy-coordinate gives sine. From those two values you can derive the remaining four trig functions using their definitions as ratios.

Worked Example

Problem: Find the exact value of csc⁡π6+tan⁡π6\csc\frac{\pi}{6} + \tan\frac{\pi}{6}.
Recall values: From the standard table, csc⁡π6=2\csc\frac{\pi}{6} = 2 and tan⁡π6=33\tan\frac{\pi}{6} = \frac{\sqrt{3}}{3}.
Add: Combine the two values.
2+33=63+33=6+332 + \frac{\sqrt{3}}{3} = \frac{6}{3} + \frac{\sqrt{3}}{3} = \frac{6 + \sqrt{3}}{3}
Answer: 6+33\dfrac{6 + \sqrt{3}}{3}

Why It Matters

The 30° angle appears constantly in physics (projectile components, force resolution) and engineering. Memorizing its exact trig values lets you solve problems on tests without a calculator and speeds up work in precalculus and calculus courses.

Common Mistakes

Mistake: Swapping the sine and cosine values for 30° and 60°.
Correction: Remember: sin⁡30°=12\sin 30° = \frac{1}{2} (the smaller value) and cos⁡30°=32\cos 30° = \frac{\sqrt{3}}{2} (the larger value). At 60° they switch.