Inverse Secant | How to Find the Exact Value

In this discussion, we are going to learn how to evaluate inverse secant.

Let us consider a few concepts:

◻️◻️◻️Concepts: ◻️◻️◻️

When working with inverse trig functions, in particularly, inverse secant,

an equation is given in the form:

y = \sec^{-1}(x)

🔷Observing the Domain:

Note that the domain of \sec^{-1}(x) is the set of all x such that y = \sec^{-1}(x) is defined.

In other words, the domain is all possible “input” values.

\sec^{-1}(x) is defined on the interval:

(-\infty, -1]\cup[1,\infty)

Thus, we say that the domain of \sec^{-1}(x) is:

(-\infty, -1]\cup[1,\infty)

Alway check to see if the x-value satisfies the domain.

If the x-value does not satisfy the domain, then there is no solution.

If the x-value is within the domain, then take the secant of both sides to get:

sec(y) =x.

🔷Converting sec(y) to cos(y)

It is much easier to evaluate cosine instead secant.

We can use the reciprocal identity

sec(y) =\frac{1}{cos(y)}.

Thus,

sec(y) =x \Rightarrow \cos(y) =\frac{1}{x}

🔷Observing the Unit Circle

Observe the unit circle, with terminal points in the form, (a,b).

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Remember that cosine takes the “a” coordinate of the terminal point.

Search for all the values of y such that cos(y) = 1/x,

where 1/x is in the “a” location of (a,b).

Then consider the range of \sec^{-1}(x) .

🔷Observing the Range

While the domain is the possible “input” values, the range is the possible “output” values.

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} .

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If you would like to know how the domain and range of inverse secant was discovered, go to this article.

🔷Choosing the Value:

Sometimes, we will find multiple values for y;

however, we must choose the value for y that satisfies the range.

Now, let us look at our example. Go to Page 2