Trigonometry
Trigonometric Equations
Basic Trigonometric Equations
tan x = a
The equation  cot x = a
Trigonometric equations
An equation that involves one or more trigonometric functions, of an unknown arc, angle or number, is called trigonometric equation.
Basic trigonometric equations
The equation  tan x = a
The solutions of the equation are the arcs of which the function value of tangent equals a.
At the points, P and P, of the unit circle, end infinite many arcs
 x = x0 + k · p = arad + k · p = arctan a + k · p,  k Î Z,
whose tangent values equals a. These arcs represent the set of the general solution (see the figure below).
While the arc x increases from - p/2 to p/2 the function tangent passes through all the values from the
range (- oo, + oo), what means that the equation tan x = a will have its basic solution within that interval.
 The basic solution x0 = arctan a    or     x0 = tan-1 a
Example:  Solve the equation,  tan x = - 2.
Solution:  Set a calculator to DEG mode and input
- and press  tan-1  (or INV tan) key    =>    x0 = - 63.43494882° = - 63°2606′′
obtained is the basic solution. To convert from decimal degree to, deg., min., sec form, press  ®DMS key.
The general solution,   x = x0 + k · 180°    =>    x - 63°2606′′ + k · 180°k Î Z,
or in radians,   x = x0 + k · p         =>    x - 1.107148718 + k · pk Î Z.
To convert from decimal degree to radians change from DEG to RAD mode by pressing DRG® key.
The equation  cotx = a
The solutions of the equation are arcs x whose function value of cotangent equals a.
At the points, P and P, of the unit circle, end infinite many arcs
 x = x0 + k · p = arad + k · p = arccot a + k · p,  k Î Z,
whose cotangent equals a. These arcs represent the general solution of the equation cot x = a.
The cotangent function passes through all the values from the range (- oo, + oo) while the arc x increases from 0 to p.
Therefore, the equation cot x = will have its basic solution within that interval, as shows the graph above.
 The basic solution is x0 = arccot a    or     x0 = cot-1 a
and we calculate it using tangent function that is,
By adding p to the formula for the negative a we compensate the difference in the definitions of the tangent of the principal values of inverse, cotangent and tangent.
We also can calculate the basic solution using the known identity  tan (p/2 - x0) = cot x0.
By substituting  cot x0 = a  or  x0 = arccot a  then,   tan (p/2 - x0) = a  or    p/2 - x0 = arctan a,
Example:  Solve the equation,  cot x = - 1.85.
Solution:  The basic solution we calculate using formula  x0 = p/2 - arctan a   or    x0 = 90° - arctan a.
Set a calculator to DEG mode and input
- 1.85  press tan-1  (or INV tan) key    =>    x0 = 90° - ( - 61.60698058° ) = 151°36′ 25.13,
and the general solution  x = x0 + k · p   or   x = x0 + k · 180° = 151°36′ 25.13 + k · 180°k Î Z.
Pre-calculus contents G