Question

Question 9 Find sin(x/2), cos(x/2) and tan(x/2) for cos x = -1/3 , x in quadrant III

Class X1 - Maths -Trigonometric Functions Page 82

Answer 1

x lies in 3rd quadrant. it means
180° < x < 270°
so, 90° < (x/2) < 135°
it means (x/2) lies on 2nd quadrant .

cosx = -1/3 = b/h
p = √(3² -1²) = ±2√2
sinx = -2√2/3 [ because x lies in 3rd quadrant ]

cosx = 2cos²(x/2) - 1 [ by formula ]
-1/3 = 2cos²(x/2) -1
2/3 = 2cos²(x/2)
1/3 = cos²(x/2)
cos(x/2) = ±1/√3
but (x/2) lies on 2nd quadrant,
so, cos(x/2) = -1/√3

cosx = 1 - 2sin²(x/2) [ by formula,
-1/3 = 1 - 2sin²(x/2)
-4/3 = -2sin²(x/2)
2/3 = sin²(x/2)
sin(x/2) = √2/√3 [ because (x/2) lies on 2nd quadrant ]

tan(x/2) = sin(x/2)/cos(x/2)
= √2/√3/(-1/√3)
= -√2