Questions: sec(x) = -3 where -π ≤ x ≤ -π/2

Transcript text: $\sec (x)=-3$ where $-\pi \leq x \leq-\frac{\pi}{2}$

Solution

Solution Steps

To solve the equation $\sec(x) = -3$ within the interval $-\pi \leq x \leq -\frac{\pi}{2}$, we need to find the angle $x$ whose secant is $-3$. Recall that $\sec(x) = \frac{1}{\cos(x)}$, so we are looking for $\cos(x) = -\frac{1}{3}$. We will use the inverse cosine function to find the angle and ensure it lies within the specified interval.

Step 1: Express $\sec(x)$ in terms of $\cos(x)$

Given the equation $\sec(x) = -3$, we can express this in terms of cosine as $\cos(x) = -\frac{1}{3}$.

Step 2: Use the inverse cosine function

To find the angle $x$, we use the inverse cosine function: \[ x = \cos^{-1}\left(-\frac{1}{3}\right) \]

Step 3: Adjust the angle to the specified interval

The inverse cosine function typically returns values in the range $[0, \pi]$. However, we need the angle $x$ to be in the interval $-\pi \leq x \leq -\frac{\pi}{2}$. Since $\cos(x)$ is negative, the angle is in the second quadrant. We adjust the angle by taking the negative of the result: \[ x = -\cos^{-1}\left(-\frac{1}{3}\right) \]

Step 4: Calculate the angle

Using the inverse cosine function, we find: \[ x \approx -1.9106 \]