How do you evaluate $sec 15$ ?

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Answer 1 · 2016-06-03T14:33:21

Exact value:

This is one of those rare questions that you can evaluate exactly using the sum and différence formulas.

First, though, let's define $sectheta$ . By the reciprocal identities $sectheta = 1/costheta$

$sec15$
$=1/cos15$

Now, $15^@$ can be written as $60^@ - 45^@$

By the sum and différence identity $cos(alpha - theta) = cosalphacostheta + sinalphasintheta$

We can therefore state the following:

$1/cos15 = 1/cos(60 - 45)$

Expanding:

$=1/(cos60cos45 + sin60sin45)$

$=1/(1/2 xx 1/sqrt(2) + sqrt(3)/2 xx 1/sqrt(2))$

$= 1/((1/(2sqrt(2)) + sqrt(3)/(2sqrt(2)))$

$= 1/((1 + sqrt(3))/(2sqrt2))$

$= (2sqrt(2))/(1 + sqrt(3))$

Rationalizing the denominator:

$= (2sqrt(2))/(1 + sqrt(3)) xx (1 - sqrt(3))/(1 - sqrt(3))$

$=(2sqrt(2) - 2sqrt(6))/-2$

$=(2(sqrt(2) - sqrt(6)))/-2$

$= sqrt6 - sqrt(2)$

Therefore, $sec15 = sqrt(6) - sqrt(2)$

Hopefully this helps!

How do you evaluate sec 15 sec 15 ?