Trigonometry
     Trigonometric Equations
      The Equations,    tan (bx + c) = m  and  cot (bx + c) = mwhere b, c and m are real numbers.
         The Equation,  cot (bx + c) = m  example
The Equations,     tan (bx + c) = m       and        cot (bx + c) = m,  where b, c and m are real numbers.
                        
The given equations can be written as F (bx + c) = m  where F substitutes a trigonometric function, x is an arc to be calculated and m is a value of a given trigonometric function.
To every trigonometric function we can determine an arc, a + k · P of which function value equals m that is   F (a + k · P) = m, where a = x0  is the basic solution, and P is the period, then
F (bx + c) = F (a + k · P)    or   b x + c = a + k · P,  thus
Thus, from obtained general solution we can write a common solutions for every given equation,
The Equation,  cot (bx + c) = m  example
Example:  Find the solutions of the equation,  cot (- 2x + 10°) - 1 = 0.
Solution:  Rearrange the given equation to the form cot (bx + c) = m, thus  cot (- 2x  +  10°) = 1,
   or   cot [- (2x  - 10°)] = 1  and since   cot (- a) = - cot a    then,   cot (2x - 10°) = - 1,
and   cot (2x - 10°) = cot (135° + k · 180°),   2x - 10° = 135° + k · 180°   =>    x = 72°30 + k · 90°.
The general solution of the equation we get direct substituting the basic solution  x0 = a  and the constant b  and c to the common solution,  b = 2c = - 10°   and  x0 = a = cot-1(-1) = 135°  give 
Pre-calculus contents G
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