How do you find the exact value of all 6 trigonometric functions for the angle pi/6?

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How do you find the exact value of all 6 trigonometric functions for the angle pi/6?

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Answer 1 · 2015-09-26T09:20:55

Construct a right angled triangle with angles $pi/6$ , $pi/3$ , $pi/2$ and look at the ratios of the lengths of the sides.

Explanation:

Consider an equilateral triangle with sides of length $2$ . It will have angles $pi/3$ , $pi/3$ , $pi/3$ .

Bisect it to make two right angled triangles, with sides of length $1$ , $sqrt(3)$ and $2$ , since $1^2 + sqrt(3)^2 = 1 + 3 = 4 = 2^2$ .

The smallest angle of one of these right angled triangles will be $pi/6$ (the others being $pi/3$ and $pi/2$ ).

Hence:

$sin (pi/6) = 1/2$ (from: sin = opposite/hypotenuse)

$cos (pi/6) = sqrt(3)/2$ (from: cos = adjacent/hypotenuse)

$tan (pi/6) = 1/sqrt(3)$ (from: tan = opposite/adjacent)

Then the reciprocal trig functions:

$csc (pi/6) = 1/sin (pi/6) = 2/1 = 2$

$sec (pi/6) = 1/cos (pi/6) = 2/sqrt(3)$

$cot (pi/6) = 1/tan (pi/6) = sqrt(3)/1 = sqrt(3)$

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How do you find the exact value of all 6 trigonometric functions for the angle pi/6?

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