◻️◻️◻️Example◻️◻️◻️
Given the following equation
y = \sec^{-1}(-2) ,
find the value for y.
Here are the steps:
🔷Observing the Domain:
Note that the domain of \sec^{-1}(x) is
(-\infty, -1]\cup[1,\infty)
Since x=-2 , the domain is satisfied.
Now take the secant of both sides to get:
sec(y) = -2.
🔷Converting sec(y) to cos(y)
Again, it is much easier to evaluate cosine instead of secant.
Thus, sec(y) = -2 can be rewritten as
cos(y) =-\frac{1}{2}
by the reciprocal identity.
🔷Observing the Unit Circle:
Looking at the unit circle, search for all the values of y such that cos(y) =-\frac{1}{2}.
Or sometimes I like to ask myself,
“Where on the unit circle does cos(y) =-\frac{1}{2}?”
On the unit circle, there are terminal points
(a,b) = (-\frac{1}{2}, \frac{\sqrt{3}}{2}) at \frac{2\pi}{3}
and terminal points
(a,b) =(-\frac{1}{2}, \frac{-\sqrt{3}}{2}) at \frac{4\pi}{3}
Remember, we observe the “a” coordinate when working with cosine.
Thus,
cos(\frac{2\pi}{3})=-\frac{1}{2}
cos(\frac{4\pi}{3})=-\frac{1}{2}
Now, consider the range of inverse secant.
The range of \sec^{-1}(x) is
(0, -\frac{\pi}{2}]\cup[\frac{\pi}{2},\pi) ,
which is the upper region of the unit circle, but not including , \frac{\pi}{2} .
🔷Choosing our Value:
Remember the y value must be within the range!
Clearly, the only value within the range is
2pi/3.
Thus, we choose \frac{2\pi}{3} .
Hence,
cos(\frac{2\pi}{3})=-\frac{1}{2}
which means that
sec(\frac{2\pi}{3})=-2
which means that
\frac{2\pi}{3}=\sec^{-1}(-2)
and thus,
y =\frac{2\pi}{3} .
We are done!
🔑 Practice is the key! If you would like more examples, practice problems with the answers, quizzes with the answers, and more regarding inverse trig functions, consider taking my Math Course on Inverse trig functions!