Question

if cos2X=1/3 and 0<=2X<=pi, find cosX

Answer 1

2X = 70.53 degrees or 1.23 radians
cos x = 0.8165

Answer 2

i need it in terms of the unit circle?

Answer 3

and an explanation would be great i havent done this in awile :/

Answer 4

double angle results

Answer 5

ok the angle 2x must be in the range 0 to pi or 0 to 180
as the cosine is positive there is only one result - that is in the first quadrant (0-90 degrees)
so 2x = 70.53 and x = 70.53/ 2
then use calculator to find cos 70.53/2

Answer 6

cos2x = cos^2-sin^2 = 2cos^2(x) -1

Answer 7

there is the formula cos 2x = 2cos^2x - 1 which can be used as well

Answer 8

therefore cos^2(x) = (1/2) ( 1+cos(2x) ) = (1/2) ( 1+ 1/3) = (1/2)(4/3) = (2/3)

therefore cos(x) = +- sqrt(2/3)

Answer 9

but 0<=2x<=pi

so 0<=x<=pi/2

which means cos is in the first quadrant , so it is positive, so take the +case.

Answer 10

\[\cos(x) = \sqrt{\frac{2}{3}}\]

Answer 11

thank you so much