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Trigonometry Angles: π/3 — Definition, Formula & Examples

π/3 is a standard angle equal to 60° whose trigonometric values appear frequently in math courses. Its exact trig ratios — derived from the 30-60-90 triangle — are values every trig student should memorize.

The angle π/3 radians corresponds to 60° in standard position. In a 30-60-90 right triangle with hypotenuse 1, the side opposite the 60° angle has length 32\frac{\sqrt{3}}{2} and the side adjacent has length 12\frac{1}{2}, giving sin⁡π3=32\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}, cos⁡π3=12\cos\frac{\pi}{3} = \frac{1}{2}, and tan⁡π3=3\tan\frac{\pi}{3} = \sqrt{3}.

Key Formula

sin⁡π3=32,cos⁡π3=12,tan⁡π3=3\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}, \quad \cos\frac{\pi}{3} = \frac{1}{2}, \quad \tan\frac{\pi}{3} = \sqrt{3}
Where:
  • π3\frac{\pi}{3} = The angle, equal to 60°
  • 32≈0.866\frac{\sqrt{3}}{2} \approx 0.866 = Sine value (and cosine of π/6)
  • 12=0.5\frac{1}{2} = 0.5 = Cosine value (and sine of π/6)

How It Works

Picture an equilateral triangle with side length 1. Cut it in half to create a 30-60-90 right triangle. The hypotenuse is 1, the short leg (opposite 30°) is 12\frac{1}{2}, and the long leg (opposite 60°) is 32\frac{\sqrt{3}}{2}. Reading ratios from the 60° vertex gives you all six trig values at π/3. On the unit circle, π/3 lands in Quadrant I at the point (12, 32)\left(\frac{1}{2},\, \frac{\sqrt{3}}{2}\right), so cosine is the xx-coordinate and sine is the yy-coordinate.

Worked Example

Problem: Find the exact value of csc(π/3) + cot(π/3).
Recall values: From the reference table:
sin⁡π3=32,cos⁡π3=12\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}, \quad \cos\frac{\pi}{3} = \frac{1}{2}
Compute reciprocals: Cosecant is the reciprocal of sine, and cotangent is cosine over sine:
csc⁡π3=23=233,cot⁡π3=1/23/2=13=33\csc\frac{\pi}{3} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}, \quad \cot\frac{\pi}{3} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
Add: Combine the two results:
233+33=333=3\frac{2\sqrt{3}}{3} + \frac{\sqrt{3}}{3} = \frac{3\sqrt{3}}{3} = \sqrt{3}
Answer: csc⁡π3+cot⁡π3=3\csc\frac{\pi}{3} + \cot\frac{\pi}{3} = \sqrt{3}

Why It Matters

π/3 shows up constantly in precalculus exams, AP Calculus integration problems, and physics whenever 60° angles arise (such as projectile motion or force decomposition). Knowing its exact values by heart lets you solve problems quickly without a calculator.

Common Mistakes

Mistake: Swapping the sine and cosine values of π/3 and π/6.
Correction: Remember: sin(π/3) = cos(π/6) = √3/2 (the larger value), and cos(π/3) = sin(π/6) = 1/2 (the smaller value). The bigger angle has the bigger sine.