Trigonometry
Trigonometric Equations
sin (bx + c) = m,  -1 <  m < 1,   cos (bx + c) = m,  -1 <  m < 1,
tan (bx + c) = m  and  cot (bx + c) = mwhere b, c and m are real numbers.
tan (bx + c) = m, example
cot (bx + c) = m, example
The Equations,  sin (bx + c) = m,  -1 <  m < 1,   cos (bx + c) = m,  -1 <  m < 1,
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  bx + c = a + k · P,  thus
Thus, from obtained general solution we can write a common solutions for every given equation,
The Equation  tan (bx + c) = m example
 Example:  Solve the equation,
Solution:  Rewrite the equation to the form tan (bx + c) = m,
We obtain the same result if we put given parameters, b, c, and corresponding basic solution
x0 = a = tan-1 m to the common solution that is, b = 1/3, c = - p/2 and x0 = a = tan-1(-Ö3/3) = - p/6
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 = 2, c = - 10°  and x0 = a = cot-1(-1) = 135°  give
Trigonometry contents B