Derive a Generalised Formula for c-d with Respect to θ

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In summary, the generalised formula for c-d with respect to θ is (c-d)=2sin(θ/2), which can be derived using trigonometric ratios and the Pythagorean theorem. It can be used for any circle as long as the values for c and d are measured on the circumference and θ is the central angle. The formula is significant because it allows for the calculation of chord length or distance between endpoints using only θ, but it has limitations for chords that do not intersect the center and for circles with irregularities.
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mfaisal
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In the above given triangle when θ = 0 then b=c
But when θ = 90 then d=0
Since d is the projection of b
How we can derive a generalised formula for d or c-d with respect to θ
Plese may kindly be elaborated
 

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Hello and welcome to MHB! (Wave)

Look at the right triangle, having side lengths $b,\,d,\,h$. Using a trigonometric function, what is the ratio of $d$ over $b$?
 

FAQ: Derive a Generalised Formula for c-d with Respect to θ

What is a generalised formula for c-d with respect to θ?

The generalised formula for c-d with respect to θ is (c-d)=2sin(θ/2), where c is the length of the chord and d is the distance between the endpoints of the chord on the circumference of a circle.

How is the generalised formula derived?

The generalised formula is derived using the relationship between the central angle, the length of the chord, and the distance between the endpoints of the chord on a circle. By using trigonometric ratios and the Pythagorean theorem, the formula can be simplified to (c-d)=2sin(θ/2).

Can the generalised formula be used for any circle?

Yes, the generalised formula can be used for any circle as long as the values for c and d are measured on the circumference of the circle and θ is the central angle formed by the chord.

What is the significance of the generalised formula?

The generalised formula is significant because it allows for the calculation of the length of a chord or the distance between the endpoints of a chord on a circle, using only the central angle θ. This can be useful in various fields such as geometry, engineering, and physics.

Are there any limitations to using the generalised formula?

One limitation of the generalised formula is that it only applies to chords that intersect the center of the circle. For chords that do not intersect the center, a different formula would need to be used. Additionally, the formula assumes a perfect circle and may not be accurate for circles with irregularities.

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