Find sin x/2, cos x/2, and tan x/2 in each of the following: cos x = - 1/3, x in quadrant III

Solution:

Since x lies in quadrant III, π < x < 3π/2

Therefore, π/2 < x/2 < 3π/4

Hence, cos x/2, and tan x/2 are negative while sin x/2 is positive as x/2 lies in quadrant II.

It is given that cos x = - 1/3

cos 2(x/2) = -1/3

By double angle formulas, 

2cos²(x/2) - 1 = - 1/3

2cos²x/2 = 1 - 1/3

cos²x/2 = (2/3) × (1/2)

cos²x/2 = 1/3

cos x/2 = ± √(1/3)

Since cos x/2 lies in quadrant II and negative so, cos x/2 = - 1/√3 (or) -√3/3

Now, sin²x/2 = 1 - cos²x/2 [Because sin²A + cos ²A = 1]

= 1 - (1/√3)²

= 1 - 1/3

= 2/3

sin x/2 = ± √(2/3)

Since sin x/2 lies in quadrant II and positive so, sin x/2 = √(2/3) (or) √6/3

Now, tan x/2 = [sin x/2] / [cos x/2]

= [-1/√3] / [√(2/3)]

= -√2

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NCERT Solutions Class 11 Maths Chapter 3 Exercise ME Question 9

Find sin x/2, cos x/2, and tan x/2 in each of the following: cos x = - 1/3, x in quadrant III

Summary:

When cos x = - 1/3, x in quadrant III, sin x/2 = √6/3, cos x/2 = -√3/3, and tan x/2 = -√2.