Answer:
[tex]\sf \dfrac{\sqrt{2}}{2}[/tex]
Step-by-step explanation:
[tex]\sf If\:\theta=\left(22\dfrac12\right)^{\circ}[/tex]
[tex]\sf \implies2\theta=2\cdot\left(22\dfrac12\right)^{\circ}=45^{\circ}[/tex]
[tex]\sf Therefore\:\sin2\theta=\sin(45)^{\circ}=\dfrac{\sqrt{2}}{2}[/tex]
Proof
If a right triangle has an interior angle of 45°, then the third interior angle will also be 45° (since the sum of the interior angles of a triangle is 180°).
This means that the two legs of the right triangle are equal in length.
Using Pythagoras' Theorem, we can state that the hypotenuse of a right triangle with two legs of equal length will be √2 times the length of a leg.
Let a = b = 1
⇒ 1² + 1² = c²
⇒ c = √2
The sine trig ratio is:
[tex]\mathsf{\sin(\theta)=\dfrac{O}{H}}[/tex]
where:
- [tex]\theta[/tex] is the angle
- O is the side opposite the angle
- H is the hypotenuse
[tex]\implies \sf \sin(45^{\circ})=\dfrac{1}{\sqrt{2} }[/tex]
[tex]\implies \sf \dfrac{1}{\sqrt{2}}\times \dfrac{\sqrt{2}}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}[/tex]
[tex]\implies \sf \sin(45^{\circ})=\dfrac{\sqrt{2}}{2}[/tex]
