Question

Solve the eqn 25 cos x = 16 sin x tan x for 0 < or = x < or = 360. Could anyone help me on this?

Answer 1

The exact answer is

`x = arctan(pm 5/4)`

with approximations `x = 51.3^circ, 231.3^circ, 308.7 ^circ` or `128.7^circ.`

Explanation:

`25 cos x = 16 sin x tan x`

`25 cos x = 16 sin x \frac{sin x}{cos x}`

`25/16 = {sin^2 x}/{cos ^2 x} = tan^2 x`

`tan x = \pm 5/4`

At this point we're supposed to do approximations. I never like that part.

`x = arctan(5/4) approx 51.3°`

`x approx 180^circ + 51.3^ circ = 231.7^circ`

`x approx -51.3^circ + 360 ^circ = 308.7 ^circ`

`or x approx 180^circ + -51.3 = 128.7^circ`

Check:

`25(cos(51.3)) - 16(sin(51.3) tan (51.3)) = -.04 quad sqrt`

`25(cos(231.3)) - 16(sin(231.3) tan (231.3)) = -.04 quad sqrt`

I'll let you check the others.