Question

Find the exact value:
cos (sin^-1 (1/2)) and explain please (:

Answer 1

So far I got sin^-1 (1/2) = cosy

Answer 2

that isn't correct, how you got that ?

Answer 3

you can first try to find sin^-1 (1/2) =.. ?

Answer 4

sin^-1 (1/2) = x
--->sin x = 1/2
for which angle is sin value = 1/2 ?

Answer 5

Oh, right!

Answer 6

π/6 or 5π/6

Answer 7

lets use pi/6
so, sin^-1 (1/2) = pi/6

cos (sin^-1 (1/2)) = cos (pi/6) =... ?

Answer 8

So wait, the inverse of sin is the same as sin?

Answer 9

is it ? ofcourse not
what we calculated was \(\sin^{-1}(1/2)=x \implies \sin x=1/2\)
why do you ask that anyways ?

Answer 10

Woops, sorry, I've been doing too much math o;

Answer 11

no problem :)
so, continue ?

Answer 12

Mhmm. (: So cos (π/6)=1/2?

Answer 13

recheck/retry ?

Answer 14

Oop, √3)/2!

Answer 15

yes, thats correct :)

Answer 16

Thank yoou! (:

Answer 17

welcoome! (:

Answer 18

Here's an alternative way to do it

First draw out a right triangle

Answer 19

Label one of the angles (that's not the 90 degree angle) as theta

Answer 20

We are given sin^-1 (1/2) as the argument for cosine

So sin(theta) = 1/2 for some theta. We really don't care what this theta is at the moment.

All we care about is that

sin(angle) = opposite/hypotenuse

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In this case, the opposite and hypotenuse are these sides here

Answer 21

so

opposite/hypotenuse = 1/2

which means

opposite = 1
hypotenuse = 2

updating the triangle to be



the missing leg of the triangle can be found through pythagoreans theorem

a^2 + b^2 = c^2

1^2 + b^2 = 2^2

1 + b^2 = 4

b^2 = 4 - 1

b^2 = 3

b = sqrt(3)