Question

If sin(O) = 24/25 and 0 is in Quadrant I then what is sin (0/2)?
Simplify your answer
​

Answer 1

`sin((\theta)/(2)) = (3)/(5)`

Explanation:

Given that:

`sin\theta = (24)/(25)`

Using trigonometric identity:

`sin^(2)x +cos^(2)x=1`

Putting
`x=\theta`

`sin^(2)\theta +cos^(2)\theta=1`

`\Rightarrow cos^2\theta = 1 - ((24)/(25))^2\\\Rightarrow cos^2\theta = (625 - 576)/(625) = (49)/(625)\\\Rightarrow cos\theta = (7)/(25)`

Positive value taken because
`\theta` is in first quadrant and value of
`cos\theta` will be positive.

Now using the formula:

`sin^2x=(1-cos2x)/(2)`

Here, putting
`2x=\theta`

`sin^2x=(1-cos2x)/(2)`

`sin^2((\theta)/(2))=(1-(7)/(25))/(2)\\\Rightarrow sin^2((\theta)/(2))=(18)/(25* 2)\\\Rightarrow sin^2((\theta)/(2))=(9)/(25)\\\Rightarrow sin((\theta)/(2))=(3)/(5)`

Positive value taken because
`\theta` is in first quadrant and value of
`sin((\theta)/(2))` will be positive.