Exact Values - Inverse Trig Expressions

Okay so this was explained horribly to me in class, so I have no clue what I am doing.

I am stuck on these problems:

Sec^-1(1/2)

Arccos(sin(π/6))

tan(Csc^-1(2^-1))

It is asking me for an exact value, however I've never gone through this before and I am only familiar with radians and degrees. Please help.

Re: Exact Values - Inverse Trig Expressions

$\displaystyle \sec x = \frac{1}{\cos x} $

I think there is a mistake in the first one

since if there is x such that $\displaystyle \sec ^{-1} \frac{1}{2} = x \Rightarrow sec x = \frac{1}{2} $

so $\displaystyle \cos x = \frac{1}{\sec x } = 2 $ and there is not any real number x satisfies that

to solve these things you should know that

$\displaystyle \sin \left(\frac{\pi}{6}\right) = \frac{1}{2} = \cos \left(\frac{\pi}{3} \right)$

$\displaystyle \cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} = \sin \left(\frac{\pi}{3}\right) $

the last one also has a problem

since if $\displaystyle \csc ^{-1} 2^{-1}= x \Rightarrow \csc x = 2^{-1} = \frac{1}{2} $

and this you give $\displaystyle \sin x = 2 $ same problem

second one if you know that sin (pi/6) equal 1/2

you have to know what is the angle where cos x = 1/2 ?

$\displaystyle \cos ^{-1} (sin \frac{\pi}{6} ) = \cos ^{-1} ( \frac{1}{2}) = $

Re: Exact Values - Inverse Trig Expressions

Again I think it's asking for an exact numerical value for Sec^-1(1/2) as well as the last one.

I'm not sure how to find exact values of inverses on the unit circle, other than take the first value which arccos or arcsin will be equal to.

Re: Exact Values - Inverse Trig Expressions

for example you know that

$\displaystyle \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} $

if they ask to find

$\displaystyle \sin ^{-1} \left( \frac{1}{2} \right) = \frac{\pi}{6} $ since the above

sin , cos is a bounded functions in real field $\displaystyle -1 \leq \sin (x),\cos(x) \leq 1 $

and I know that $\displaystyle \sec x = \frac{1}{\cos x } $

so when they ask to find

$\displaystyle \sec ^{-1} \left(\frac{1}{2}\right) $

I begin to let it equal to x then taking the sec function to both sides

$\displaystyle \sec ^{-1} \left(\frac{1}{2}\right) = x $

$\displaystyle \sec \left( \sec ^{-1} \left(\frac{1}{2}\right) \right) = \sec (x) $

but sec is the inverse of sec^-1 and I know that

$\displaystyle f(f^{-1}(x)) = x $

so you will have on the left hand side

$\displaystyle \frac{1}{2} = \sec (x) $

I know that $\displaystyle \sec (x) = \frac{1}{\cos (x) } $

$\displaystyle \frac{1}{2} = \frac{1}{\cos (x) } $

$\displaystyle \cos (x) = 2 $

and there is not any real number like that you can check this here

i means it is imaginary number not real