Now that sec, csc and cot have graphs and domains of their own, put them to work — solving equations, including ones with transformed angles, by converting straight back to sine, cosine and tangent.
9 min read3 worked examples
The strategy: convert, then solve as usual
Every equation involving $\sec$, $\csc$ or $\cot$ can be turned into an equation involving $\cos$, $\sin$ or $\tan$ by taking a reciprocal of both sides. Once you've done that, it's a standard trig equation: find the principal value, then use the symmetry of the graph (or a CAST diagram) to pick out every solution in the given range.
The core rearrangements
$\sec\theta = k \iff \cos\theta = \dfrac1k$. $\csc\theta = k \iff \sin\theta = \dfrac1k$. $\cot\theta = k \iff \tan\theta = \dfrac1k$.
This only breaks down if $k=0$, since $\dfrac1k$ would be undefined — but that's fine, because $\sec\theta$, $\csc\theta$ and $\cot\theta$ can never actually equal zero (they only ever take values with $|y|\ge1$, or any real value for $\cot$, but never exactly $0/1$ in a way that causes a problem — the one genuine exception is $\cot\theta=0$, which happens when $\cos\theta=0$, and is perfectly fine to convert: $\tan\theta$ is undefined there instead, so you'd spot it immediately).
Worked example 1
A direct reciprocal equation
Solve $\sec\theta = -2$ for $0 \le \theta < 360°$.
MethodSubstitute, solve for the new variable, then unwind the transformation
1
Let $x = 2\theta-60°$. Since $0\le\theta<180°$: $0\le2\theta<360°$, so $-60°\le x<300°$.
Apply the scaling and shift to the range in the same order they were applied to $\theta$.
2
$\cot x = 1 \Rightarrow \tan x = 1$.
Take the reciprocal.
3
Principal value $x=45°$; tangent has period $180°$, so $x = 45° + 180°n$ for integer $n$.
Tan repeats every 180°, unlike sin and cos which repeat every 360°.
4
Testing values in $-60°\le x<300°$: $n=0$ gives $x=45°$; $n=1$ gives $x=225°$. ($n=-1$ gives $x=-135°$, outside the range.)
Only keep values of x that fall inside the widened range from step 1.
5
Unwind: $\theta = \dfrac{x+60°}{2}$, so $\theta = 52.5°$ or $\theta = 142.5°$.
Reverse the substitution $x=2\theta-60°$ to return to $\theta$.
▪ solved
Solutions: $\theta = 52.5°$ or $\theta = 142.5°$.
Common mistake
Forgetting to widen the range for a substituted variable is the single most common error here. If $\theta$ ranges over $360°$ but the equation involves $2\theta$, the new variable ranges over $720°$ — twice as many solutions can appear before you divide back down.
Check yourself
Solve $\sec\theta = 1.8$ for $0 \le \theta < 360°$, giving your answers to 1 decimal place. (Hint: convert to cosine first, then decide which quadrants give a positive value.)