How do you find exact value of $sin (\frac{π}{3})$ $sin (pi/ 3)$ ?

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How do you find exact value of $sin (\frac{π}{3})$ $sin (pi/ 3)$ ?

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Answer 1 · 2016-07-24T03:04:37

$sin (\frac{π}{3}) = \frac{\sqrt{3}}{2}$ $sin (pi/3) = sqrt(3)/2$

Explanation:

$\frac{π}{3}$ $pi/3$ Radians $= 60$ $= 60$ Degrees

$sin 60 = \frac{\sqrt{3}}{2}$ $sin 60 = sqrt(3)/2$

To prove this consider an equilateral trigangle ABC of side 2 units as diagram below.

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The bisector of angle A meets side BC at D which is the midpoint of BC.

By Pythagoras:
$A B^{2} = A D^{2} + D B^{2}$ $AB^2 = AD^2 + DB^2$
$2^{2} = A D^{2} + 1^{2} \to A D = \sqrt{4 - 1} = \sqrt{3}$ $2^2 = AD^2 +1^2 -> AD = sqrt(4-1) = sqrt(3)$

Angle B = 60 Degrees
Therefore: $sin (60) = \frac{A D}{A B} = \frac{\sqrt{3}}{2}$ $sin(60) = (AD)/(AB) = sqrt(3)/2$

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