Answer EFG only Thanks In this exercise you will use the dis

Answer E,F,G only. Thanks

In this exercise, you will use the distance formula and the Pythagorean identity. a) Give the coordinates of P and Q in terms of theta. b) Use the distance formula to show that (PQ)^2 = 2 - 2 cos (theta/2) cos theta - 2 sin (theta/2) sin theta and (PR)^2 = 2 -2cos(theta/2) c) Explain why the distances PQ and PR have the same length. d) Use the result in part (c), equate (PQ)^2=(PR)^2, show that sin(theta/2) sin theta = cos (theta/2) (1 - cos theta). e) Use the result in part (d), show that sin^2(theta/2)(2 - 2costheta)= (1 - cos theta. f) Solve the equation in part (e) for the quantity sin(theta/2). g) Explain why you can safely divide both sides by 1 - cos x in part (f).

Solution

e)

sin *sin /2 = (cos /2) (1 - cos )

2*(sin /2)* (cos /2) *sin /2 = (cos /2) (1 - cos )

2*(sin /2) *sin /2 = (1 - cos )

2*(sin /2)^2 = (1 - cos )

MULTIPLY BOTH SIDES BY (1 - cos )

2*(sin /2)^2 *(1 - cos )= (1 - cos )^2

(sin /2)^2 *(2 -2 cos )= (1 - cos )^2

f)

(sin /2)^2 *(2 -2 cos )= (1 - cos )^2

divide BOTH SIDES BY (1 - cos ) since , is element of 0 to pi/4

2*(sin /2)^2 = (1 - cos )

2*(sin /2)^2 = (1 – ( 1- 2 *sin /2)^2) )

2*(sin /2)^2 = (2 *sin /2)^2)

It is true for 0 to pi/4

g)

since , is element of 0 to pi/4 ,

  (1 - cos ) will not be 0

Hence we can divide by   (1 - cos )