if cos x=2/3 and x is in quadrant 4 find tan(x/2),sin(x/2),and cos(x/2)

i got
sin(x/2)=squr 1/6
tan(x/2)=squr 2/6
cos(x/2)=squr 5/6

To find the values of sin(x/2), tan(x/2), and cos(x/2), we can use the half-angle formulas.

Given that cos(x) = 2/3 and x is in quadrant 4, we know that cos(x) is positive in quadrant 4. Since cos(x) = adjacent/hypotenuse, we can assign the adjacent side of the triangle as 2 and the hypotenuse as 3. Using the Pythagorean theorem, we can find the opposite side as √(3^2 - 2^2) = √5.

Now we have the values of the sides in the triangle: adjacent = 2, opposite = √5, and hypotenuse = 3.

To find sin(x/2), we can use the formula:
sin(x/2) = ±√((1 - cos(x))/2)

Since x is in quadrant 4, x/2 will also be in quadrant 4, where sin(x/2) is positive.

Plugging in the values, we get:
sin(x/2) = √((1 - cos(x))/2)
= √((1 - 2/3)/2)
= √(1/6)
= √(1/6)
= √6/6
= √6/6

Therefore, sin(x/2) = √6/6.

To find tan(x/2), we can use the formula:
tan(x/2) = sin(x)/(1 + cos(x))

Plugging in the values, we get:
tan(x/2) = sin(x)/(1 + cos(x))
= (√6/6)/(1 + 2/3)
= (√6/6)/(5/3)
= (√6/6) * (3/5)
= √6/10

Therefore, tan(x/2) = √6/10.

To find cos(x/2), we can use the formula:
cos(x/2) = ±√((1 + cos(x))/2)

Since x is in quadrant 4, x/2 will be in quadrant 2, where cos(x/2) is negative.

Plugging in the values, we get:
cos(x/2) = -√((1 + cos(x))/2)
= -√((1 + 2/3)/2)
= -√(5/6)
= -√5/√6
= -√5/√6 * (√6/√6)
= -√5√6/√6 * √6/√6
= -√30/√6
= -√30/√(6*6)
= -√30/√36
= -√30/6

Therefore, cos(x/2) = -√30/6.

To find the values of sin(x/2), tan(x/2), and cos(x/2), given that cos(x) = 2/3 and x is in quadrant 4, you can use the half-angle identities.

1. First, find the value of cos(x/2):
The half-angle identity for cosine is:
cos(x/2) = ±√((1 + cos(x))/2)

Since x is in quadrant 4, where cosine is positive, we can use the positive square root:
cos(x/2) = √((1 + cos(x))/2)

Substituting the value of cos(x) = 2/3:
cos(x/2) = √((1 + 2/3)/2) = √((5/3)/2) = √(5/6) = √5/√6 = √5/6

2. Next, find the value of sin(x/2):
The half-angle identity for sine is:
sin(x/2) = ±√((1 - cos(x))/2)

Since x is in quadrant 4, where sine is negative, we will use the negative square root:
sin(x/2) = -√((1 - cos(x))/2)

Substituting the value of cos(x) = 2/3:
sin(x/2) = -√((1 - 2/3)/2) = -√((1/3)/2) = -√(1/6) = -1/√6 = -√6/6

3. Finally, find the value of tan(x/2):
The half-angle identity for tangent is:
tan(x/2) = sin(x/2)/cos(x/2)

Substituting the values we found earlier:
tan(x/2) = (-√6/6)/(√5/6) = -√6/√5 = -√(6/5) = -√6/√5

Therefore, the values are:
sin(x/2) = -√6/6
tan(x/2) = -√6/√5
cos(x/2) = √5/6

I mentally draw the triangle

adjacent x:2
hypotenuse:3
opposite: -sqrt5
Half angle formlas:
sin(x/2)=+- sqrt((1-cosx)/2)
it is in the 4th quad, so neg sign applies
sin(x/2)=- sqrt(1/6)
see http://www.freemathhelp.com/images/halfangles.png for the other formulas (or see your text)