Question

Trigonometry question

Answer 1

Explanation:

`\sin( ( \alpha )/(2) ) = \sqrt{ (1 - \cos( \alpha ) )/(2) }`

So let find cos

We know that

`\tan {}^(2) (x) + 1 = \sec {}^(2) (x)`

We know the value of tan

`( (8)/(5) ) {}^(2) + 1 = \sec {}^(2) (x)`

`(64)/(25) + 1 = \sec {}^(2) (x)`

`(89)/(25) = \sec {}^(2) (x)`

`( √(89) )/(5) = \sec(x)`

Sec is the reciprocal of cosine so cosine is

`\cos( \alpha ) = (5)/( √(89) )`

Which becomes

`\cos( \alpha ) = (5 √(89) )/(89)`

So since we know cos a, let's find sin

`\sqrt{ (1 - (5 √(89) )/(89) )/(2) }`

`\sqrt{ (89 - 5 √(89) )/(2) }`