given that x is in quadrant 1 and that sin x = 2/3, find exact values for:

Question

is this correct?

1) sec (30° + x) = -3/√15

2) cos (1/2 x) = √(-√5)/2

Answer 1

nope, how are you doing these ?

1. the first one I would do this way ...
if sinx = 2/3, then cosx = √5/3

sec(30+x) = 1/cos(30+x)
= 1/(cos30cosx - sin30sinx)
= 1/((√3/2)(√5/3) - (1/2)(2/3) )
= 1/( (√15 - 2)/6)
= 6/(√15 - 2) ---- my calculator confirms this answer

#2, in your answer you are taking the √ of a negative number. That should have been your first clue that you were wrong.

I would use the double angle formula
cos 2A = 2cos^2 A - 1
and cosx = 2cos^2 (x/2)1 - 1
√5/3 = 2cos^2 (x/2) - 1
cos^2 (x/2) = (√5/3 + 1)/2 = (√5 + 3)/6

cos x/2 = √( √5 + 3) )/√6

again, my calculator says this answer is correct