Finding time difference between two arriving wave fronts

[Callumnc1]

Homework Statement:

Please see below

Relevant Equations:

S = Rθ

For part(b),

The solution is,

However, where did they get the formula shown in red from?

Many thanks!

 

[kuruman]

$AB=2R\sin30^o.$ Do you see why? Hint: Draw a perpendicular from the center to the base AB. It splits AB into two equal parts. What is the length of each part?

 

[Callumnc1]

$AB=2R\sin30^o.$ Do you see why? Hint: Draw a perpendicular from the center to the base AB. It splits AB into two equal parts. What is the length of each part?

Thanks I see it now @kuruman ! Is the reason why they used radians instead of degrees in the arc length formula is because radians is a ratio of the length while degrees is not.

Many thanks!

 

[kuruman]

What you call the arc length formula is actually the definition of the angle as the ratio of the arc length to the radius. As such it has no dimensions.

 

[Callumnc1]

Thanks @kuruman ! But isn't degrees have no dimensions too?

Many thanks!

 

[haruspex]

Thanks @kuruman ! But isn't degrees have no dimensions too?

Many thanks!

Yes, both are dimensionless in the usual dimensionality system; they just have different units.
That said, many have found this unsatisfactory and have attempted to develop a dimension for angles. If interested, see https://www.physicsforums.com/insights/can-angles-assigned-dimension/

 

[Callumnc1]

Ok thank you @haruspex ! - I will check that out.

Many thanks!