QuestionSolve for $x$ in the equation $a \sec x = 3$ .

Studdy Solution

STEP 1

Assumptions1. The equation given is $a \sec x=3$ . We need to solve for $x$ , assuming $a$ is a constant and $x$ is in radians3. $\sec x$ is the reciprocal of $\cos x$ , i.e., $\sec x = \frac{1}{\cos x}$

STEP 2

First, we rewrite the equation in terms of $\cos x$ .
$a \sec x= \Rightarrow a \frac{1}{\cos x} =$

STEP 3

Next, we solve for $\cos x$ .
$\cos x = \frac{a}{3}$

STEP 4

Now, we need to find the value of $x$ such that $\cos x = \frac{a}{3}$ . We use the inverse cosine function, $\cos^{-1}$ , to do this.
$x = \cos^{-1}\left(\frac{a}{3}\right)$ This is the general solution for $x$ . However, it's important to note that the solution for $x$ depends on the value of $a$ and the domain of the cosine function. The cosine function is defined for all real numbers, but its range is $-1 \leq \cos x \leq1$ . Therefore, for a real solution to exist, $-3 \leq a \leq3$ .

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