Question

If csc ( x ) = 4, for 90 ∘ < x < 180, then sin ( x /2 ) = ? cos ( x /2 ) = ? tan ( x/ 2 ) = ?

Answer 1

see below

Explanation:

cscx=4

sinx=1/4

cosx=-√(1-sin^2x)=-√(1-(1/16))=-√(15/16)=-√15/4

note: cos<0,sin>0 in quadrant II

sin(x/2)=
`\sqrt{(1+cos(x))/(2) } =\sqrt\frac{1+\sqrt{(15)/(4) } }{2} =\sqrt{(4+√(15) )/(8) }`

cos(x/2)=
`\sqrt{(1-cos(x))/(2) } =\sqrt{\frac{1-\sqrt{(15)/(4) } }{2} } =\sqrt{(4-√(15) )/(8) }`

tan(x/2)=
`(sin(x))/(1-cos(x)) =((1)/(4) )/((1-(√(15) )/(4) )) =((1)/(4) )/((4-√(15) )/(4) ) =(1)/(4-√(15) )`

Calculator check:

`sinx=(1)/(4)`

`x\\eq 165.52`˚ in quadrant II

`(x)/(2) \\eq 82.76` ˚

sin(x/2)≈sin(82.76)≈0.9920..

exact value=√(4+√15)/8≈0.9920..

..

cos(x/2)≈cos(82.76)≈0.1260..

exact value=√(4-√15)/8≈0.1260..

..

tan(x/2)=tan(82.76)≈7.872..

exact value=1/(4-√15)≈7.872..