Solve tan(x) + 3 cot(x) - 5 sec(x) = 0 where 0<x<180°

Answer 1

Convert all factors into their equivalent sin(x) or cos(x) forms.

sin(x)/cos(x) + 3cos(x)/sin(x) - 5/cos(x) = 0. Multiply all terms by cos(x)

sin(x) + 3cos²(x) - 5 = 0 Sub in 1-sin²x for the cos²(x) [Pythagorean Theorem]

sin(x) + 3(1-sin²x) - 5 = 0 expand and reorganize a bit

3sin²x - sin(x) + 2 = 0

Now, if you were solving this like a polynomial - say 3x² - x +2 = 0 you would come up with these factors:

(3x-2)(x-1) = 0

So instead of x we have sinx

(3sinx-2)(sinx-1) = 0. By the zero product rule, either one of those terms could be 0 because the only way you get zero by multiplying two things together if is one of them iz 0..so

either

sinx-1= 0

sinx = 1 or x = pi/2 (90 degrees)

or

sinx(x) = 2/3

so angle is about 41.8 degrees.