Answer:
[tex]sin(\dfrac{\theta}{2}) = \dfrac{3}{5}[/tex]
Step-by-step explanation:
Given that:
[tex]sin\theta = \dfrac{24}{25}[/tex]
Using trigonometric identity:
[tex]sin^{2}x +cos^{2}x=1[/tex]
Putting [tex]x=\theta[/tex]
[tex]sin^{2}\theta +cos^{2}\theta=1[/tex]
[tex]\Rightarrow cos^2\theta = 1 - (\dfrac{24}{25})^2\\\Rightarrow cos^2\theta = \dfrac{625 - 576}{625} = \dfrac{49}{625}\\\Rightarrow cos\theta = \dfrac{7}{25}[/tex]
Positive value taken because [tex]\theta[/tex] is in first quadrant and value of [tex]cos\theta[/tex] will be positive.
Now using the formula:
[tex]sin^2x=\dfrac{1-cos2x}{2}[/tex]
Here, putting [tex]2x=\theta[/tex]
[tex]sin^2x=\dfrac{1-cos2x}{2}[/tex]
[tex]sin^2(\dfrac{\theta}{2})=\dfrac{1-\dfrac{7}{25}}{2}\\\Rightarrow sin^2(\dfrac{\theta}{2})=\dfrac{18}{25\times 2}\\\Rightarrow sin^2(\dfrac{\theta}{2})=\dfrac{9}{25}\\\Rightarrow sin(\dfrac{\theta}{2})=\dfrac{3}{5}[/tex]
Positive value taken because [tex]\theta[/tex] is in first quadrant and value of [tex]sin(\dfrac{\theta}{2})[/tex] will be positive.
