find the exact value of the expression.
1. cos^-1 (-√ 3/2)
2. tan^-1 (-1)
3. sin(tan^-1 2)
4. cos (sin ^-1 3/5)

Question

find the exact value of the expression.
1. cos^-1 (-√ 3/2)
2. tan^-1 (-1)
3. sin(tan^-1 2)
4. cos (sin ^-1 3/5)

jdoe0001 · Answer

well, get  your Unit Circle :), and that'd show you who is who

jdoe0001 · Answer

1st is usually a given in most unit circles

the 2nd one, tan = y/x, so where "y" and "x" have the same value, they'd give 1, and when either is negative, they'd give -1

jdoe0001 · Answer

the 3rd one

you'd firstl need to find the angle WHOSE tangent is 2, or WHOSE y/x is 2, is a positive 2, so,"y" and "x" are either both positive or both negative

jdoe0001 · Answer

the 3rd one and 4th one, don't seem to be any of the "regular usual" angles in most unit circles, so, seem you may end up using your calculator for those

anonymous · Answer

so what does the inverse have to do with this?

jdoe0001 · Answer

is not an inverse, is a misnomer, or miswriting per se

(cos/sin/tan)$\large ^{-1}$   simply means

what is the angle WHOSE cos/sin/tan is THIS NUMBER?

jdoe0001 · Answer

so an easy one, number 2)

$tan^{-1}(1)$

what is the angle WHOSE tangent is 1?

well, taking a peek at my unit circle, $45^o$ has a sine and cosine which are exact, value and sign

so, 2nd quadrant they differ in sign
so 3rd quadrant, they're the same in sign, I check the "reference angle" for 45, that is 180+45, low and behold, they're both the same value, so

$\cfrac{same}{same}$ = 1

so, those are the angles WHOSE tangent is 1

anonymous · Answer

the questions actually says tan^-1 (-1) but does that change anything?

jdoe0001 · Answer

well $\cfrac{same}{-same}\ \ or \ \ \cfrac{-same}{same} \implies -1$

anonymous · Answer

so then it would be either 7pi/4 or 3pi/4 correct?

jdoe0001 · Answer

yes, it'd be both

anonymous · Answer

ok so my problem is, the answer to this one problem is supposed to be -pi/4...

jdoe0001 · Answer

well, no

jdoe0001 · Answer

the negative exponential in trig identities, doesn't stand for any reciprocal :/, unfortunately, is a misnomer, it really just mean

what is the angle whose function gives this value?

say
cos(x)   = 1234567
so, if I were to ask, what is the value whose cosine is "1234567"? or
$\color{red}{cos^{-1}(1234567)}?$

well, since the cos(x)   = 1234567, so the angle would be "x"