Given Sec(x)= 3/2 with 3pi/2 < x < 2pi, find sin (x/2).

Question

need the answer to this problem.

Answer 1

3pi/2 < x < 2pi

Sec(x) = Hypotenuse/Adjacent side.

Sec(x) = 3/2

So hypotenuse = 3 and Adjacent side = 2.

Then opposite side is √5

Sin(x) = opposite side/hypotenuse = -√5/3.

Comments

The value of the trigonometric function sin (x/2) is - 1/√6.

Answer 2

The value of the trigonometric function sec(x) is 3/2, 3/π2 < x < 2π  that means angle x lies in IV quadrant.

From the Double - Angle formulas, one may generate easily the Half - Angle formulas.

cos (x) = cos² (x/2) - sin² (x/2) = 2cos² (x/2) - 1 = 1 - 2sin² (x/2).

⇒ cos (x) = 1 - 2sin² (x/2)  → (1)

Using the reciprocal identity cos (x) = 1/sec (x).

The value of cos (x) = 1/(3/2) = 2/3.

To find the value of sin (x/2), substitute the value cos (x) = 2/3 in eq (1).

2/3 = 1 - 2sin² (x/2)

2sin² (x/2) = 1 - 2/3

2sin² (x/2) = (3 - 2)/3 = 1/3

sin² (x/2) = 1/(3 * 2) = 1/6

sin (x/2) = ± √(1/6)

sin (x/2) = ± 1/√6.

Since the angle x lies in IV quadrant, the value of the trigonometric function sin (x/2) is - 1/√6.