Question

"After showing that `2sectheta-tantheta=3` can be written as `sqrt10cos(theta-18.4)=2`, solve for `theta` (`0<theta<360`)". I proved the first part, can somebody please solve for `theta`?

Answer 1

Solve 2sec x - tan x = 3

Ans: `69^@23 and 327^@63`

Explanation:

I solve it another way.
`2/(cos x) - sin x/(cos x) = 3`
2 - sin x = 3cos x
sin x + 3cos x = 2.(1)
Call `tan a = sin a/(cos a) = 3 = tan 71.57` --> cos a = 0.316
Equation (1) -->
`sin x + (sin a)/(cos a)cos x = 2`
sinx.cos a + sin a.cos x = 2cos a = 0.632
sin (x + a) = sin (x + 71.57) = 0.632. Two solutions:
a. (x + 71.57) = 39.20 --> x = 39.20 - 71.57 = - 32.37 (or `327^@63`)
b. x + 71.57 = 180 - 39.20 = 140.80.
x = 140.80 - 71.57 = `69^@23`
Check by calculator.
x = 69.23 --> 2sec = 5.64 --> tan x = 2.64.
2sec - tan x = 5.64 - 2.64 = 3. OK
x = - 32.37 --> 2sec = 2.37 --> tan x = - 0.63
2sec - tan x = 2.37 + 0.63 = 3. OK