Unit circle problem

Jan 2020
3
0
Sweden
Hello guys, so I have this problem in my assignment that I just can't seem to understand and answer - hope you guess are able to help, thank you.

a) In the diagram, radii drawn to endpoints of a chord of the unit circle determine a central angle a. Show that the length of the chord is L = sqrt(2-2cos(Ø))

b) By using the substitution Ø = a/2 in the double angle formula cos2Ø = 1-2 sin^2 Ø, derive a formula for sin a/2, that is a half-angle formula for the sine function.

c) Use the result in (a) and your result in (b) to show that the length of the chord is L = 2 sin(Ø/2)
 
Dec 2014
165
125
USA
(a) diagram look like this, maybe? If so, you can use the law of cosines to show the chord length is $L = \sqrt{2-2\cos{\phi}}$

unit_circle_chord.jpg
(b) do the substitution as directed and solve for $\sin\left(\dfrac{a}{2}\right)$ ... give it a try

Try (b) and we'll see what can be done for (c)
 
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Plato

MHF Helper
Aug 2006
22,506
8,663
Hello guys, so I have this problem in my assignment that I just can't seem to understand and answer - hope you guess are able to help, thank you.

a) In the diagram, radii drawn to endpoints of a chord of the unit circle determine a central angle a. Show that the length of the chord is L = sqrt(2-2cos(Ø))

b) By using the substitution Ø = a/2 in the double angle formula cos2Ø = 1-2 sin^2 Ø, derive a formula for sin a/2, that is a half-angle formula for the sine function.

c) Use the result in (a) and your result in (b) to show that the length of the chord is L = 2 sin(Ø/2)
You can find this page most useful.
 
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