Basic Trigonometric Equations
Basic trigonometric equations involve equations where the unknown variable x appears as the argument of one or more trigonometric functions.
A straightforward example of a trigonometric equation is:
$$ \sin x = c $$
where c is any real number, and x is the variable we’re solving for.
Solving Basic Trigonometric Equations
Each type of trigonometric equation has its own method for finding solutions.
| Equation | Solution |
|---|---|
| $$ \sin x = c $$ | This equation has solutions only if -1 ≤ c ≤ 1; otherwise, there’s no solution. The solutions are given by $$ x = \alpha + 2\pi k \vee (\pi - \alpha) + 2\pi k $$ See proof and explanation. |
| $$ \cos x = c $$ | This equation has solutions only if -1 ≤ c ≤ 1; otherwise, it’s unsolvable. The solutions are $$ x = \alpha + 2\pi k \vee -\alpha + 2\pi k $$ See proof and explanation. |
| $$ \tan x = c $$ | This equation is solvable for any real value of c. The solutions are $$ x = \alpha + k \pi $$ See proof and explanation. |
| $$ \sin x = \sin y $$ | This equation is true when the angles are either equal, \(x = y\), or supplementary, \(x + y = \pi\), with an integer multiple of 2π added. $$ x = y + 2k \pi \ ∨ \ x+y = \pi + 2k \pi $$ See example. |
| $$ \cos x = \cos y $$ | This equation is satisfied when the angles are either equal, \(x = y\), or opposite, \(x = -y\), with any integer multiple of 2π. $$ x = y + 2k \pi \ ∨ \ x = -y + 2k \pi $$ See example. |
| $$ \tan x = \tan y $$ | This equation holds when the angles are congruent, \(x = y\), plus any integer multiple of π. $$ x = y + k \pi $$ See example. |
| $$ \cot x = \cot y $$ | To solve this equation, rewrite the cotangent in terms of tangent using the related angle \(\pi/2 - x\) and \(\pi/2 - y\): $$ \frac{\pi}{2}-x=\frac{\pi}{2}-y+k \pi $$ See example. |
| $$ \sin x = - \sin y $$ | This case can be rephrased as sin x = sin (-y) because sine is an odd function. $$ x = y + 2k \pi \ ∨ \ x+y = \pi + 2k \pi $$ See example. |
| $$ \sin x = \cos y $$ | This equation can be simplified by writing it as sin x = sin (π/2 - y) with cos(y) = sin(π/2 - y). $$ x = (\pi/2 - y) + 2k \pi \ ∨ \ x + (\pi/2 - y) = \pi + 2k \pi $$ See example. |
| $$ \sin x = - \cos y $$ | This equation reduces to sin x = sin (y - π/2) by rewriting -cos(y) as sin(y - π/2) since sine is an odd function. $$ x = (y - \pi/2) + 2k \pi \ ∨ \ x + (y - \pi/2) = \pi + 2k \pi $$ See example. |
| $$ \cos x = - \cos y $$ | This equation simplifies to cos x = cos (π - y) since supplementary angles have opposite cosines. $$ x = (\pi - y) + 2k \pi \ ∨ \ x = - (\pi - y) + 2k \pi $$ See example. |
| $$ \tan x = - \tan y $$ | This equation can be rewritten as tan x = tan (-y) because tangent is an odd function: -tan(y) = tan(-y). $$ x = (-y) + k \pi $$ See example. |
And so forth.
