Question

sec ^-1(x^2-1) (5marks)

Solution

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4 (306 Votos)

Diego Mestre · Tutor por 5 anos

Resposta

To find the value of \( \sec^{-1}\left(x^{2}-1\right) \), we need to understand the properties of the secant function and its inverse.The secant function, denoted as

, is the reciprocal of the cosine function. It is defined as:

The inverse of the secant function, denoted as

, is the function that returns the angle whose secant is a given value. In other words, if \( \sec^{-1}(y) = \theta \), then \( \sec(\theta) = y \).Now, let's consider the expression \( \sec^{-1}\left(x^{2}-1\right) \). We need to find the angle whose secant is equal to

.To do this, we can use the fact that the secant function is periodic with a period of

. This means that if \( \sec(\theta) = y \), then \( \sec(\theta + 2k\pi) = y \) for any integer

.Therefore, we can write:

where

is the angle whose secant is equal to

and

is an integer.To find the value of

, we need to consider the range of the secant function. The secant function is defined for all real numbers except for angles where the cosine function is zero. Therefore, the range of the secant function is all real numbers except for

, where

is an integer.Since the secant function is periodic with a period of

, we can restrict the range of

to be within the interval

without loss of generality.Now, we need to find the angle

whose secant is equal to

. To do this, we can use the fact that the secant function is the reciprocal of the cosine function. Therefore, we can write:

Now, we need to find the angle

whose cosine is equal to

. This can be done by using the inverse cosine function, denoted as

.Therefore, we can write:

So, the value of \( \sec^{-1}\left(x^{2}-1\right) \) is equal to \( \cos^{-1}\left(\frac{1}{x^{2}-1}\right) \).Note: It is important to mention that the value of

must be such that

is not equal to 0 or

, as these values would make the secant function undefined.