Trigonometric equations

A trigonometric equation is an equation in which the unknown appears only inside a trigonometric ratio — for instance $\sin\theta=\tfrac12$ or $2\cos^2x-7\cos x+3=0$. This is a fundamentally different object from a trigonometric identity (such as $\sin^2\theta+\cos^2\theta=1$): an identity holds for every admissible value of the angle, while an equation holds only for particular values, which is exactly what we are asked to find.

Because every trigonometric ratio repeats itself after a fixed period ($2\pi$ for sine and cosine, $\pi$ for tangent and cotangent), a trigonometric equation essentially never has just one answer. If one value $\theta_0$ satisfies $\sin\theta=k$, so does $\theta_0+2\pi$, $\theta_0+4\pi$, $\theta_0-2\pi,\ldots$ — the whole family generated by the period. A trigonometric equation can also have no solution at all: since $\sin\theta$ and $\cos\theta$ never leave $[-1,1]$, an equation like $\sin\theta=\tfrac32$ is simply unsatisfiable, no matter how far we search.

Principal solution vs. general solution

Because a trigonometric equation's solution set is periodic, it is useful to separate two ideas.

• The general solution is the complete infinite family of angles satisfying the equation, expressed with an integer parameter (traditionally $n$) that sweeps through $\mathbb Z$ and regenerates every valid angle via the function's periodicity.
• The principal solution is a single representative angle from that family — specifically, the one of smallest absolute value lying in $[-\pi,\pi]$. (The interval $[0,2\pi]$ works just as well for this purpose and gives an equally valid representative; the two conventions simply parametrise the same infinite solution set starting from a different anchor point, as Example 3.43's comparison shows.) A trigonometric equation can have up to two candidates in $[-\pi,\pi]$ (one from each of two quadrants where the ratio takes that value); when that happens, we always keep the numerically smaller one as the principal solution.

Restricting each ratio to a fixed window is also exactly what lets us later talk about the inverse sine, cosine or tangent. The three windows used throughout this section are:

For a ratio expressed via a reciprocal function (cosecant, secant, cotangent), we first flip it to sine, cosine or tangent and then read the principal value off the matching window above — e.g. $\operatorname{cosec}\theta=-2$ is read as $\sin\theta=-\tfrac12$, and $\cot\theta=\sqrt3$ is read as $\tan\theta=\tfrac1{\sqrt3}$.

Building the three general-solution formulas

Rather than memorising the final formulas as arbitrary rules, it helps to see why each one takes its particular shape — the shape comes directly from turning a "ratio equals ratio" equation into a product that vanishes.

For $\sin\theta=k$ ($-1\le k\le1$): let $\alpha$ be the numerically smallest angle with $\sin\alpha=k$. Writing the equation as $\sin\theta-\sin\alpha=0$ and applying the sum-to-product identity turns it into a product of a cosine and a sine factor, each of which can independently be set to zero. Solving those two zero-conditions separately and then recombining the two resulting families (one running over even multiples of $\pi$, the other over odd multiples, shifted by $\pm\alpha$) collapses neatly into the single alternating-sign formula

The alternating $(-1)^n$ is what lets one formula sweep out solutions from both the quadrant where sine equals $k$ directly and its "mirror" quadrant — for even $n$ it reproduces $\alpha$ shifted by a full period $n\pi\to$ effectively $2\pi$-blocks, for odd $n$ it reproduces the supplementary-angle branch.

For $\cos\theta=k$ ($-1\le k\le1$): let $\alpha\in[0,\pi]$ satisfy $\cos\alpha=k$. The same idea — write $\cos\theta-\cos\alpha=0$, convert to a product via the sum-to-product identity, and set each factor to zero — this time produces two families that recombine into

Here the symmetry is a plain $\pm$ rather than an alternating sign, because cosine is an even function: $\cos(-\alpha)=\cos\alpha$, so both $+\alpha$ and $-\alpha$ (equivalently $2\pi-\alpha$) are always simultaneously valid.

For $\tan\theta=k$ (any real $k$): let $\alpha\in\left(-\tfrac\pi2,\tfrac\pi2\right)$ satisfy $\tan\alpha=k$. Clearing denominators in $\dfrac{\sin\theta}{\cos\theta}=\dfrac{\sin\alpha}{\cos\alpha}$ gives $\sin\theta\cos\alpha-\cos\theta\sin\alpha=0$, i.e. $\sin(\theta-\alpha)=0$, whose zeros are simply every integer multiple of $\pi$ added to $\alpha$:

Tangent's period is only $\pi$ (half of sine/cosine's), which is exactly why its general solution needs just one un-alternated family instead of two interleaved ones.

The combined form $a\cos\theta + b\sin\theta = c$

Many equations mix a cosine term and a sine term of the same angle on one side of an equation set equal to a constant. The trick is to manufacture a single auxiliary angle $\alpha$ so that the left side collapses into one cosine (or sine) of a shifted angle:

Put $a=r\cos\alpha$ and $b=r\sin\alpha$, where $r=\sqrt{a^2+b^2}$ (so $\alpha$ is simply the polar angle of the point $(a,b)$, and $\tan\alpha=\tfrac ba$). Substituting,

The original equation becomes $r\cos(\theta-\alpha)=c$, i.e. $\cos(\theta-\alpha)=\dfrac{c}{r}=\dfrac{c}{\sqrt{a^2+b^2}}$ — an ordinary $\cos(\cdot)=$ constant equation in the single unknown $\theta-\alpha$, solvable by the cosine formula above. Writing $\dfrac{c}{\sqrt{a^2+b^2}}=\cos\varphi$ for a convenient $\varphi$, the general solution is

In practice one does not need to memorise $\alpha=\tan^{-1}(b/a)$ symbolically — it is far quicker to just divide the whole equation by $r=\sqrt{a^2+b^2}$ and recognise the resulting coefficients as $\sin$ or $\cos$ of a familiar angle (as Examples 3.54 does with $r=2$, recognising $\tfrac{\sqrt3}{2}$ and $\tfrac12$ as $\sin\tfrac\pi3$ and $\cos\tfrac\pi3$).

Summary of general solutions

How the worked examples put this to use

Examples 3.42–3.55 progressively layer these formulas onto increasingly disguised equations.

• Finding principal solutions directly (Example 3.42): each ratio is matched to a known reference angle, its sign is used to pin down which of the two allowed quadrants applies, and the window table above hands back a single numerically-smallest angle — including reciprocal-ratio cases like $\operatorname{cosec}\theta=-2$, first flipped to $\sin\theta=-\tfrac12$.
• Reading off a general solution once the equation is already in "ratio = ratio" form (Examples 3.43, 3.44): the matching row of the summary table is applied immediately once a reference angle $\alpha$ in the right window has been identified. …