Solving Reciprocal Trig Equations

For $\sec x=k$ with $|k|\ge1$, solve $\cos x=1/k$. Cosecant converts to sine in the same way. If $\cot x=k\ne0$, use $\tan x=1/k$; if $k=0$, use cosine zero instead.

An inverse-trig key returns one principal value, not all solutions. Sine gives an angle from $-90^\circ$ to $90^\circ$, cosine from $0^\circ$ to $180^\circ$, and tangent strictly between $-90^\circ$ and $90^\circ$. Use symmetry and the period to find the rest.

In radians, if $\beta$ is the calculator value, sine solutions are $\beta+2n\pi$ and $\pi-\beta+2n\pi$; cosine solutions are $\pm\beta+2n\pi$; tangent solutions are $\beta+n\pi$. Here $n$ is any integer. Replace $2\pi,\pi$ by $360^\circ,180^\circ$ in degrees. Keep only values inside the interval and list repeated values once.

When the input contains a shift or a multiplier, name the whole input $u$. Apply that same change to both interval endpoints. Solve for $u$ there, then undo the change. This prevents missing a repeated cycle.