Question

How do you evaluate `sec(cos^-1(1/2))` without a calculator?

Answer 1

`sec(cos^(-1)(1/2)) = 2`

Explanation:

Another way, without calculating `cos^(-1)(1/2)`:

Consider a right triangle with an angle `theta = cos^(-1)(1/2)`. Then `cos(theta) = 1/2`, meaning the ratio of its adjacent side to the hypotenuse is `1/2`. Thus the ratio of the hypotenuse to its adjacent side, that is, `sec(theta)`, is `2/1 = 2`.

Thus `sec(cos^(-1)(1/2)) = sec(theta) = 2`

Note that this same reasoning shows that in general, `sec(cos^(-1)(x)) = 1/x`

Answer 2

`2`

Explanation:

Understand that `color(blue)(cos^-1(1/2)=cos(theta)=(1/2)`

As we know that, cosine function is the reciprocal of secant function,

`color(brown)(sec(theta)=1/(cos(theta))`

`rarrsec(theta)=1/(1/2)`

`rArrsec(theta)=2`

Hope this helps... :)