Thread: Exact Value

I have to evaluate the exact value of tan(15).

To give some context, I have to find the area of a dodecagon, and I used the uber super cool formula:
ns^2/(4tan(180/n)) where n is the number of sides and s is the side length.

Thanks.

Originally Posted by DivideBy0

I have to evaluate the exact value of tan(15).

To give some context, I have to find the area of a dodecagon, and I used the uber super cool formula:
ns^2/(4tan(180/n)) where n is the number of sides and s is the side length.

Thanks.

Use a half-angle formula.

I can never remember the formula for tan(a/2), so I do it this way:

sin(a/2) = (+/-)sqrt{1 - cos(a)}/sqrt{2}
cos(a/2) = (+/-)sqrt{1 + cos(a)}/sqrt{2}
(where we get the + or - from which quadrant the angle is in.)

So
tan(a/2) = sin(a/2)/cos(a/2) = (+/-)sqrt{1 - cos(a)}/sqrt{1 + cos(a)}

In this case, a = 30:
tan(15) = sqrt{1 - cos(30)}/sqrt{1 + cos(30)}

= sqrt{1 - sqrt{3}/2}/sqrt{1 + sqrt{3}/2}

= sqrt{2 - sqrt{3}}/sqrt{2 + sqrt{3}}

You will want to rationalize this, so multiply the numerator and denominator by sqrt{2 - sqrt{3}}. I'll just give you the answer:

tan(15) = sqrt{7 - 4sqrt{3}}

-Dan

Thanks m8!

Originally Posted by DivideBy0

Thanks m8!

I appreciate the thanks ( ), but what the heck is "m8?"

-Dan

Originally Posted by topsquark

I appreciate the thanks ( ), but what the heck is "m8?"

-Dan

It's a 'leet' abbreviation of 'mate' (used commonly in australia)

Originally Posted by DivideBy0

It's a 'leet' abbreviation of 'mate' (used commonly in australia)

Ah! Obvious once it's explained.

-Dan