Piano Guidance
Photo by Teona Swift Pexels Logo Photo: Teona Swift

What is the chord of a circle called?

A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle.

en.wikipedia.org - Chord (geometry) - Wikipedia
What sound did Van Halen use for Jump?
What sound did Van Halen use for Jump?

Oberheim OB-Xa The synth which Van Halen used in “Jump” is an Oberheim OB-Xa. This machine uses polyphonic, substractive synthesis to generate its...

Read More »
Last updated Jul 7, 2021
What is the easiest piano?
What is the easiest piano?

Here are seven easy piano songs for beginners to get you started. Twinkle Twinkle. Twinkle Twinkle Little Star is always popular, especially with...

Read More »
Last updated Dec 8, 2019

Geometric line segment whose endpoints both lie on the curve

This article is about the line segment defined on a curve. For other uses, see Chord (disambiguation) A chord of a circle is a straight line segment whose endpoints both lie on a circular arc. The infinite line extension of a chord is a secant line, or just secant. More generally, a chord is a line segment joining two points on any curve, for instance, an ellipse. A chord that passes through a circle's center point is the circle's diameter. The word chord is from the Latin chorda meaning bowstring.

BX is a chord

(as is the diameter segment AB). The red segmentis a(as is the diameter segment).

In circles [ edit ]

Among properties of chords of a circle are the following:

Chords are equidistant from the center if and only if their lengths are equal. Equal chords are subtended by equal angles from the center of the circle. A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle. If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem).

In conics [ edit ]

The midpoints of a set of parallel chords of a conic are collinear (midpoint theorem for conics).[1]

In trigonometry [ edit ]

Chords were used extensively in the early development of trigonometry. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the chord function for every 7+1/2 degrees. In the second century AD, Ptolemy of Alexandria compiled a more extensive table of chords in his book on astronomy, giving the value of the chord for angles ranging from 1/2 to 180 degrees by increments of 1/2 degree. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part.[2] The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle. The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure). The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be (cos θ, sin θ), and then using the Pythagorean theorem to calculate the chord length:[2] crd ⁡ θ = ( 1 − cos ⁡ θ ) 2 + sin 2 ⁡ θ = 2 − 2 cos ⁡ θ = 2 sin ⁡ ( θ 2 ) . {displaystyle operatorname {crd} heta ={sqrt {(1-cos heta )^{2}+sin ^{2} heta }}={sqrt {2-2cos heta }}=2sin left({frac { heta }{2}} ight).}

What piano should a beginner use?
What piano should a beginner use?

While 88-key digital pianos are the best choice for students planning on learning to play traditional piano, students can learn to play with a...

Read More »
Last updated Dec 16, 2020
How difficult is it to learn piano?
How difficult is it to learn piano?

The piano is one of the most difficult and rewarding instruments to learn; not only do you have to learn to read notes and translate them to the...

Read More »
Last updated Sep 11, 2022

The last step uses the half-angle formula. Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve-volume work on chords, all now lost, so presumably, a great deal was known about them. In the table below (where c is the chord length, and D the diameter of the circle) the chord function can be shown to satisfy many identities analogous to well-known modern ones: Name Sine-based Chord-based Pythagorean sin 2 ⁡ θ + cos 2 ⁡ θ = 1 {displaystyle sin ^{2} heta +cos ^{2} heta =1,} crd 2 ⁡ θ + crd 2 ⁡ ( π − θ ) = 4 {displaystyle operatorname {crd} ^{2} heta +operatorname {crd} ^{2}(pi - heta )=4,} Half-angle sin ⁡ θ 2 = ± 1 − cos ⁡ θ 2 {displaystyle sin {frac { heta }{2}}=pm {sqrt {frac {1-cos heta }{2}}},} crd ⁡ θ 2 = 2 − crd ⁡ ( π − θ ) {displaystyle operatorname {crd} {frac { heta }{2}}={sqrt {2-operatorname {crd} (pi - heta )}},} Apothem (a) c = 2 r 2 − a 2 {displaystyle c=2{sqrt {r^{2}-a^{2}}}} c = D 2 − 4 a 2 {displaystyle c={sqrt {D^{2}-4a^{2}}}} Angle (θ) c = 2 r sin ⁡ ( θ 2 ) {displaystyle c=2rsin left({frac { heta }{2}} ight)} c = D 2 crd ⁡ θ {displaystyle c={frac {D}{2}}operatorname {crd} heta }

The inverse function exists as well:[3]

θ = 2 arcsin ⁡ c 2 r {displaystyle heta =2arcsin {frac {c}{2r}}}

See also [ edit ]

References [ edit ]

en.wikipedia.org - Chord (geometry) - Wikipedia
Do you tip for baseball lessons?
Do you tip for baseball lessons?

If you are meeting for lessons at a baseball facility where the coach works and gives other lessons then I would not tip him unless he does...

Read More »
Last updated May 15, 2022
What key is best for sad songs?
What key is best for sad songs?

The minor scale is the pattern in western music typically associated with sad feelings. It includes three different variations called the natural...

Read More »
Last updated Nov 20, 2021
Why are musicians smarter?
Why are musicians smarter?

A 1996 study found musicians to be measurably smarter than non-musicians because musicians practice skills like concentration and memory more than...

Read More »
Last updated Nov 12, 2020
Why is piano the best instrument?
Why is piano the best instrument?

Pianos cover all 88 notes of the musical scale, unlike other instruments, offering an incredible, unparalleled range. Pianos go higher and lower in...

Read More »
Last updated Jul 10, 2018