Elementary Trigonometric Equations
Elementary trigonometric equations are equations in which the unknown variable \(x\) appears inside one or more trigonometric functions. Solving these equations means finding all the values of \(x\) that make the equation true.
A typical example is:
$$ \sin x = c $$
where \(c\) is a real constant and \(x\) is the unknown variable.
How to Solve Elementary Trigonometric Equations
Most trigonometric equations can be solved by exploiting the periodicity and symmetry of trigonometric functions. The table below summarizes the most common elementary forms and their general solutions.
| Equation | General solution |
|---|---|
| $$ \sin x = c $$ | Real solutions exist only when \(-1 \le c \le 1\). Let \(\alpha = \arcsin c\). Then: $$ x = \alpha + 2\pi k \vee x = \pi - \alpha + 2\pi k $$ See the proof and explanation. |
| $$ \cos x = c $$ | Real solutions exist only when \(-1 \le c \le 1\). Let \(\alpha = \arccos c\). Then: $$ x = \alpha + 2\pi k \vee x = -\alpha + 2\pi k $$ See the proof and explanation. |
| $$ \tan x = c $$ | This equation has real solutions for every value of \(c\). Let \(\alpha = \arctan c\). Then: $$ x = \alpha + k\pi $$ See the proof and explanation. |
| $$ \sin x = \sin y $$ | Two angles have the same sine when they are equal or supplementary, apart from whole rotations: $$ x = y + 2k\pi \vee x = \pi - y + 2k\pi $$ See the example. |
| $$ \cos x = \cos y $$ | Two angles have the same cosine when they are equal or opposite, apart from whole rotations: $$ x = y + 2k\pi \vee x = -y + 2k\pi $$ See the example. |
| $$ \tan x = \tan y $$ | Two angles have the same tangent when they differ by an integer multiple of \(\pi\): $$ x = y + k\pi $$ See the example. |
| $$ \cot x = \cot y $$ | Since cotangent has period \(\pi\), the solutions are: $$ x = y + k\pi $$ See the example. |
| $$ \sin x = -\sin y $$ | Because sine is an odd function, \(-\sin y = \sin(-y)\). Therefore: $$ x = -y + 2k\pi \vee x = \pi + y + 2k\pi $$ See the example. |
| $$ \sin x = \cos y $$ | Using the identity $$ \cos y = \sin\left(\frac{\pi}{2}-y\right), $$ the equation becomes \(\sin x = \sin\left(\frac{\pi}{2}-y\right)\). Hence: $$ x = \frac{\pi}{2}-y+2k\pi \vee x = \frac{\pi}{2}+y+2k\pi $$ See the example. |
| $$ \sin x = -\cos y $$ | Since $$ -\cos y = \sin\left(y-\frac{\pi}{2}\right), $$ the equation can be rewritten as \(\sin x = \sin\left(y-\frac{\pi}{2}\right)\). Therefore: $$ x = y-\frac{\pi}{2}+2k\pi \vee x = \frac{3\pi}{2}-y+2k\pi $$ See the example. |
| $$ \cos x = -\cos y $$ | Since $$ -\cos y = \cos(\pi-y), $$ the equation becomes \(\cos x = \cos(\pi-y)\). Hence: $$ x = \pi-y+2k\pi \vee x = y-\pi+2k\pi $$ See the example. |
| $$ \tan x = -\tan y $$ | Because tangent is an odd function, $$ -\tan y = \tan(-y), $$ so: $$ x = -y+k\pi $$ See the example. |
In all formulas, \(k \in \mathbb{Z}\).
These elementary cases form the foundation for solving more complex trigonometric equations. In many situations, a complicated equation can be transformed into one of these standard forms by applying trigonometric identities, symmetry properties, or algebraic manipulations.
