Question

Question #f1faf

Answer 1

`color(red)2.625`

Explanation:

`sec 292.5 = sec (585/2) = 1/cos(585/2)`

Let us calculate `cos (585/2)`

`cos (585/2) = cos [(720-135)/2] = cos(360 - 135/2)`

`= cos(-135/2) = cos(135/2)`. ----------(1.)

Now,
`cos (x/2) = +-sqrt[(1+cosx)/2]`

`therefore cos(135/2) = +-sqrt[(1+cos135)/2]`

`cos 135 = cos (180 - 45) = -cos 45 = -1/sqrt2 = -0.71`

`=> cos(135/2) = +-sqrt[(1-0.71)/2]`

`= +- sqrt(0.29/2) = +-sqrt(.0145) = +-0.381`

but `135/2 = 67.5 < 90.` Hence `cos(135/2)` will be positive.

`therefore cos(135/2) = 0.381`

From (1.) `cos (585/2) = cos(135/2) = 0.381`

`=> sec 292.5 = sec (585/2) = 1/cos(585/2) = 1/.381 = color(red)2.625`

If you need help with the half angle formula, do comment below.