Question

if tan x = 3÷4 wherex lies in third quadrant then find sin (x/2 ) , cos ( x/2) and tan (x/2)​

Answer 1

Answer:

Formulas used

1. sin x/2 =√1-cos x/2

2.cos x/2=√1+ cos x /2

tan x=3/4 ,means it lies in 3 rd quadrant so ,

sin and cos both are negative .

then find cos by the triangle

Answer 2

Answer:

sin x/2=3√2 /5

cos x/2 =-√7/5  and tan x/2= -3√(2/7)

Step-by-step explanation:

tan x = 3/4

x is in 3rd quadrant  so x=180+ y  ( y<90)

x/2=90+y/2 ( y/2 < 45)

So x/2 is in 2nd quadrant  Now sec x= √(1+tan²x=√(1+9/16)

=√(16+9)/16= -√(25/16)

= -5/4  ( as sec x in 3rd quadrant is -ve)

Cos x=-4/5

Now cos x/2= √(2cos²x-1)

=√ ( 2*16/25-1) = - √{ (32-25)/25 - 1 }= -√(7/25)=-√7/5

So cos x/2 =-√7/5

now sin x/2 =√ (1-cos²x/2) =√1-7/25

=√(18/25)=3√2 /5 ( as sin x/2 in 2nd quadrant is +ve)

sin x/2=3√2 /5

tan x/2= sin x/2 / cos x/2 = (3√2 /5) / ( √7/5)

tan x/2= -3√ (2/7)