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.

The reference angle, the acute angle to the horizontal axis, is $30^\circ$. Sine is negative in quadrants III and IV.

The calculator's $-30^\circ$ is outside the interval. Both retained angles have sine $-1/2$.

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.