Question

If cos x = 2 / 3 and x is in quadrant 4, find:

a. sin(x / 2)

b. cos(x / 2)

c. tan(x / 2)

Answer 1

A

Explanation:

cos(x)=2/3 in Q 4

sin(x/2)=+√(1-cos(x))/2

√(1-cos(x))/2=√(1-[2/3]/2=√(1/3)/2=-√(1/6) because sin is negative in Q 4

Answer 2

See below.

Explanation:

Because cos x = 2/3 the adjacent side = 2 and hypotenuse = 3 so the length of the opposite side =

√(3^2 - 2^2) = -√5 (its negative because we are in Quadrant 4).

So sin x = -√5/3.

(a) sin (x /2) = - √ [ (1 - cos x)/2 ]

= -√(1 - 2/3)/ 2)

= -√(1/6). or -0.4082.

(b) cos (x/2) = √ [ (1 + cos x)/2]

= √ 5/6 or 0.9129.

(c) tan (x /2) = ( 1 - cos x) / sin x.

= ( 1 - 2/3) / -√5/3

= -0.4472.