Find the exact value of sinx/2 if cosx = 2/3 and 270 < x < 360.
A)1/3
B)-1/3
C)sqrt 6/6
D)-sqrt 6/6
C, since I KNOW cosx is always positive but I don't know the work involved. I know the half angle formula
First of all, x/2 will be in the second quadrant, since x is in the fourth quadrant. The sine of x/2 will therefore be positive.
Use the formula for sin (x/2) in terms of cos x.
sin(x/2) = sqrt([1-cos(x)]/2) = sqrt (1/6) = sqrt6/6
You got the right answer, but you it ssmes to have been a lucky guess.
Cos x is NOT always positive, but it is in this case.
Thank you. It wasn't really a guess it was either C or D and then I just knew it was positive so that just leaves C.
Jon,
Perhaps it would help if you drew an x-y axis system with a unit radius vector in each of the four quadrants.
then in quadrant 1
sin T = y/1 so +
cos T = x/1 so -
tan T = y/x so +
then in quadrant 2
sin T = y/1 so +
cos T = x/1 so - because x is - in q 2
tan T = y/x so -
then in quadrant 3
sin T = y/1 so -
cos T = x/1 so -
tan T = y/x so + because top and bottom both -
then in quadrant 4
sin T = y/1 so -
cos T = x/1 so +
tan T = y/x so -
sin has same sign as its inverse csc
cos has same sign as its inverse sec
tan has same sign as its inverse ctan
then in quadrant 1
sin T = y/1 so +
cos T = x/1 so +
tan T = y/x so +
To find the exact value of sin(x/2), given that cos(x) = 2/3 and 270 < x < 360, you can use the half-angle formula for sine.
The half-angle formula for sine is:
sin(x/2) = ± √((1 - cos(x))/2)
Since x lies in the fourth quadrant (270 < x < 360), the sine is negative. Therefore, we will choose the negative sign in the formula.
First, calculate the value of cos(x/2):
cos(x/2) = ± √((1 + cos(x))/2)
cos(x/2) = ± √((1 + 2/3)/2)
cos(x/2) = ± √(5/6)
Now, we can use the Pythagorean identity sin^2(x/2) + cos^2(x/2) = 1 to find sin(x/2):
sin(x/2) = -√(1 - cos^2(x/2))
sin(x/2) = -√(1 - (5/6))
sin(x/2) = -√(1/6)
sin(x/2) = -√(6/6)
sin(x/2) = -√6/√6
sin(x/2) = -√6/6
Therefore, the exact value of sin(x/2) is -√6/6, which corresponds to option D) in the given choices.